The formation of Madeira island is usually attributed to a hotspot.
What is a hotspot? – A hotspot is a region where the mantle beneath is more hot than elsewhere which can generate volcanic activity at the surface of the earth where great magma effusions can build large volcano edifices. It is thought that an hotspot is the result of a plume in the mantle (an upwelling of abnormally hot rock within the Earth’s mantle rising from the core). If the hotspot is located at the sea it can lead to the formation of islands.
Global volcanic activity distribution
Mainly at plate tectonic boundaries – At the surface of the earth the volcanic activity is mainly concentrated along the boundaries of tectonic plates. That’s what happen for example in the Azores archipelago where magma ascends through the mid-Atlantic Ridge and nearby fractures.
Volcanic activity at hotspots – However, when a volcanic island is not located at the boundaries but is instead located on the middle of a tectonic plate its formation is generally attributed to a hotspot.
There, in the middle of the Pacific Plate, there are a group of islands thought to have formed by the same hotspot. Their alignment is considered to be an evidence of the hotspot that was in its origin. As you can see in the image the hotspot kept still, and as the Pacific Plate moved above, several islands were formed, aligned at the surface, forming a “hotspot track”.
In the following image you can see an animated model of the formation of islands by a mantle plume as the plate moves over it:
The hotspot of Madeira?
Generally it has been assumed that Madeira Island originated from a upwelling mantle plume. The majority of authors consider that an hotspot formed the Madeira Island. But was it?
Yes, Madeira island is a shield volcano, an island formed after the effusion of a large amount of magma that leaked from the mantle. And not far from Madeira island there are several seamounts, considerable in size, thought to be of volcanic origin as well. And some attempts were made to align Madeira Islands with those underwater mountains in a way compatible with an hotspot origin… But is there really a hotspot track like the one we can see at Hawaii?
I think some years ago it would be acceptable that scholars attempted to find a hotspot alignment there and even came with some possible hotspot tracks. But nowadays I think it does not make much sense anymore. Why? Because now there are much more clear images of the bottom of the ocean. Currently, anyone can check at Google Earth if there is really any kind of alignment between Madeira and other bathymetric abnormalities that could suggest a hotspot.
This is a possible alignment that has been presented by some geologists:
Is this picture what we get when we check current imagery of the bottom of the ocean? No, I don’t think so.
Check this Google Earth image:
Yes, there are some obvious alignments. But those alignments look more like faults, it would be very hard to see there an alignment of islands forming a hotspot “track”.
Madeira island formed along faults on the oceanic crust
What the bottom of the ocean tell us? Well I think it is pretty obvious that the picture we get from Google Earth is of a system of faults along which some seamounts/islands formed. Madeira island was formed along one of this faults too.
There is a group of faults oriented roughly NE-SW and another approximately W-E. Madeira Island formed along a W-E fault.
If this interpretation is correct Madeira Island was not formed as a result of a hotspot but rather was formed as a result of magma leaking to the surface through tectonic faults.
Age of Madeira Island
Volcanism in Madeira Island (and Desertas islands) began about more than 5 Ma ago (in the Miocene), until about 0,007 Ma (Pleistocene). The island of Porto Santo is much older dating back to about 14 million years ago.
The oceanic crust around these islands is very old, around 140-180 million years old. This is the region where the Atlantic Ocean first began to open. The oldest oceanic crust that can be found on earth has an age of 180 million years, even in the Pacific.
Building of Madeira island edifice
Madeira Island is a shield volcano. It began to form on the ocean floor at a depth of 4000 m. Whether it originated from a hotspot or not, magma kept leaking from below the crust building an underwater edifice. After reaching the surface of the water, about 5 Ma ago, the volcano kept growing until reaching an altitude of more than 1862 m (the current altitude of Pico Ruivo).
Near São Vicente, at Sítio dos Lameiros, we can find marine fossils (corals, sea snails, echinoderms, etc), not at sea level but at an altitude of more than 400 m above sea level. It is believed that when the top of the volcano reached the surface of the water some coral reefs formed on it. Later, somehow the island uplifted and these coral structures ended up on a higher altitude (see animation above). The exact process by which this happened is still not well understood and has yet to be proper researched and explained.
When did the faults on the oceanic crust formed?
Some authors suggest that the Madeira-Tore Rise was the result of a lithospheric buckling process . But if it was so, why would the buckling process produce such a perfect straight line? Anyone can perceive straight lines in the Google Earth image referred before, not only in the Madeira-Tore Rise (the western NE-SW line) but there are other straight lines parallel to that one, and other lines making a specific angle with that major alignment… Did the buckling occurred as a result of compression (or transpression) along previous existent faults?
In the Paleozoic, on the Variscan Orogeny, this region of the globe was subjected to great stresses, producing several major faults (for example in portuguese mainland territory, Verin-Regua fault, Vilariça fault, and Messejana fault, are all faults that were formed on the Variscan Orogeny ). These faults were later reactivated both in the Mesozoic (in the rifting of opening of the Atlantic Ocean) and in the Cenozoic (when the region began to be subjected again to compressive forces in the Alpine Orogeny).
Some major faults might have also been formed when the Atlantic Ocean began to open. Even the Messejana fault is considered to be part of the so-called Central Atlantic Magmatic Province :
The Messejana fault was intruded by a dike of basalt around 200 Ma, by the time the Atlantic was beginning to open. About the Messejana fault Pais et al.  say it is a:
“major crustal fault that has had a complex tectonic history as an initial Late Variscan sinistral strike-slip fault, later reactivated as a transtensional fault allowing the emplacement of Triassic-Jurassic magmas, and again reactivated as a transtensional sinistral strike-slip fault during the Cenozoic.”
What is curious is that the major faults oriented NE-SW (that we can observe in Google Earth image, in Madeira region), have the same orientation of the major faults that we can find depicted in the northernmost sector of the Central Atlantic Magmatic Province. Check the previous image.
In the following image you can see how those faults have the same orientaion of the Messejana fault:
This NE-SW orientation is the orientation of the coast of Africa, it is the orientation of the great rifting fault along which in this region the land broke off, along which the Atlantic Ocean began to open.
What does this mean? It means that maybe the major faults that we can found today on the bottom of the ocean, around the islands of Madeira, have also originated when the Atlantic Ocean began to open, about 200 Ma ago. If this is true the intraplate islands and seamounts around Madeira have probably a tectonic origin, an origin related to magma leaking along active faults in the Mesozoic and Cenozoic. And so a mantle plume origin does not seem to fit so well anymore…
Madeira island is aligned W-E
But Madeira island is not aligned along a NE-SW fault. It is aligned along a W-E fault.
These W-E faults can be interpreted as conjugated faults. These W-E faults form an angle of about 60º with the major NE-SW faults. This is compatible with Riedel conjugates. Faults oriented W-E can be R’ faults and the faults oriented NE-SW can be R faults formed along the great Rift of the opening of the North Atlantic Ocean:
The directions of the strike-slip movements depicted in the image are not the current movement directions that can be detected in the Madeira region now. However, the movements could have been those depicted in the image at the time of the opening of the Ocean. The opening of the ocean might have been a dextral strike-slip (see image below ) which is compatible with the dextral movements depicted above:
Some authors refer dextral movements in NE-SW faults. For example Ellero et al. refers a “framework of conjugated fault systems with the ENE-WSW dextral shear zones developed during the main Variscan phase” .
Later, by Mesozoic and Cenozoic, the kinematics on these structures suffered an inversion. The initial dextral movement of the major faults (NE-SW) became sinistral, and the sinistral movement of the conjugated faults (W-E) became dextral. Oceanic faults would also be affected, changing its movement direction.
Faults on Madeira Island itself
There are abundant tectonic lineaments in the island. However the most important fault in Madeira is one that runs along an approximately W-E direction and a dextral movement has been detected in it . According to Fonseca et al. :
“The most important lineament (Seixal-Machico lineament – SML) locally with an expressive geomorphological degraded scarp, is running parallel to the elongation of the island which has the maximum extension of 56 km. It corresponds to a major dextral fault confirmed by some field observations.” 
Note that the configuration of the island and the orientation of this main fault fits in the major W-E fault that we can perceive in the imagery of the bottom of the ocean.
Cenozoic faults – Alpine compression
Another possibility is that the faults in the oceanic crust are not so old. They could have also formed in the context of the Alpine Orogeny. During convergence of the African and European plates, during the Cenozoic, some variscan faults were reactivated and some new faults were formed. This might have affected the oceanic crust.
The same Alpine N-S compression that originated mountain ranges alternated with basins (oriented ENE-WSW), both on Iberia and in the north of Africa, might have affected the ocean crust as well. Faults generated on the continents might have propagated to the oceanic crust. Faults along the Atlas range, along the Betic orogeny or along the western Iberia might have propagated to the oceanic crust generating there conjugated faults.
The case of Canary Islands
In the case of Madeira, the faults on the oceanic crust might have propagated from continental tectonic structures like the Nazaré fault or Messejana fault ( a land-ocean continuity can be established). However Madeira is far from the continent, so Canaries are commonly used as a better example of propagation faults. Canariries are close to Africa, and in that region of Africa there is a great fault, the Atlas fault. Some authors have proposed that the islands of Canaries formed along a fault that have propagated from the Atlas tectonic chain. Critics argue that there is no evidence of continuous faults connecting the two areas .
In the case of Canary islands what I notice is that there are two alignments that are compatible with the two set of alignments that we can find in oceanic crust around Madeira. There is a clear W-E alignment of some islands while the rest of the islands align in a NE-SW direction. Do Canary islands formed also along ancient faults which were originated in the early stages of the opening of the Ocean? Even the Selvagens Island and NE seamounts seem to fit in these pattern of two sets of directions:
Now you may ask, how would a hostpot track on Canary Islands look like?
The hotspot of Canary Islands
Remember that if the plume concept is correct, all hotspot tracks on a single, moving plate should have a similar configuration (assuming the plate behaves like a rigid body). In the image below you can see that an effort was made by the author to make the configuration of the Canary hotspot track similar to the Madeira one. But, as you can see it is far from fitting to the the actual alignment of the Canary islands:
Of course the author argue that there might be anisotropies in the oceanic crust, and that the proximity to the continent might have affected the alignment of the islands . Personally what I see is that the hotspot model don’t fit in the case of Canary Islands as well.
As we have seen before the supposed hostspot track of Madeira does not seem to fit to what the seafloor tell us. And as we can see now the supposed Madeira and Canarias hotspot tracks does not seem to fit together either.
If what we see at the bottom of the ocean is really a system of faults, if faults are more relevant than hipothetic hotspots to explain the origin of intraplate islands like Madeira and Canarias, then it would be important to know how the faults were behaving by the time the islands formed. Madeira formed about 5 Ma ago, Porto Santo Island about 14 Ma ago, Selvagens Islands about 27 Ma, and Canarias about 25 Ma ago. All of these islands are recent, geologically speaking. All of them were formed in the Cenozoic under the influence of the Alpine Orogeny. We know that in the Cenozoic the strike slip movement in the Azores-Gibraltar fault has been dextral. We also know that the major fault in Madeira Island is dextral. Faults in Atlas tectonic chain are sinistral in the NE-SW direction, and dextral in the High Atlas region. Other NE-SW faults are sinistral under the Alpine convergence.
Below is my attempt to set the faults kinematics based on the known tectonic movements. The known movements are yellow and the inferred movements are red.
In Madeira Island the observed fault movement is compatible with what we would expect from other known movement of faults in the region of the north Atlantic. It is for example compatible with the dextral movement of the most important tectonic fault in the region, which is the boundary between Africa and Euroasia plates (Azores-Gibraltar fault). Assuming that Madeira is connected to that boundary by a system of conjugated faults we would expect that the movement in Madeira would be dextral, and in fact it is.
It would be interesting to check in loco in the different islands (and if possible in the seafloor), if these kind of movements can be found, if there are evidence of movements that match the movements inferred here.
I think that scientific research on alternatives to the hotspot/plume model should be encouraged for the sake of a better understanding of the origin of intraplate islands, like Madeira.
I am not a specialist in this area, but I am curious enough to find myself interested in this subject to the point of reading a few scientific papers and to try to find clues to what makes sense to me. Even without being an expert, and under the risk of committing some errors here, I don’t find the hotspot explanation very convincing (specially in the case of Madeira), and I am not the only one…
There are already some scientists that are questioning the existence of mantle plumes to explain the origin of islands. For example in the following video Prof Gillian Foulger of the Durham University questions the hotspot of Iceland:
In the portuguese scientific community it seems that alternatives to the hotspot model are already being considered. For example in the Notícia Explicativa da carta Geológica da Madeira , which is the written document that explains the most recent geological map of Madeira (from 2010), the authors consider that:
“Although the nature and origin of these relief forms are not fully undestood, beyond the model of the existence of a mantle plume and a hotspot track, we consider the possibility that these structures are the geomorphic expression of the tectonic deformation and volcanic activity, involving two systems of regional fracturing expressed by propagated faults or by leaky faults.”
 Ramalho et al. (2005). Tectonics of Ponta de São Lourenço, Madeira Island. PDF
 J. Pais et al. (2012). The Paleogene and Neogene of Western Iberia (Portugal). PDF
 Cebriá et al. (2002). Geochemistry of the Early Jurassic Messejana–Plasencia dyke (Portugal–Spain); Implications on the Origin of the Central Atlantic Magmatic Province. PDF
 Ellero, A. et al. (2012). Structural Geological Analysis of the High Atlas (Morocco): Evidences of a Transpressional Fold-Thrust Belt. PDF
 Fonseca, P. et al. (1998). Tectonic lineaments from Madeira island evidenced from Satellite Image analysis and preliminary geological data. PDF
 Viñuela, J. (2007). The Canary Islands Hot Spot. PDF
 Carracedo, J. and Torrado, F. (2013). Geological and Geodynamic Context
of the Teide Volcanic Complex. PDF
 Geldmacher, J. (2005). New 40Ar/39Ar age and geochemical data from seamounts
in the Canary and Madeira volcanic provinces: Support for the mantle plume hypothesis. PDF
 Silveira, A. et al. (2010). Notícia Explicativa da Carta Geológica da Ilha da Madeira, na escala 1:50000. PDF