I don't know about Walker, but Penrose most emphatically does believe that it is electrochemical. In fact, he believes to have pinpointed the crucial trick mechanism as a quantum electrodynamical effect owing to a particular molecular arrangement in microtubules. In any case, his ideas are simply hypotheses, and don't count as evidence for or against anything. But make no mistake: Penrose is a frank and unabashed materialist when it comes to consciousness.
Personally, I think Penrose is wrong. First, I didn't accept the arguments he laid out in The Emperor's New Mind that consciousness cannot be algorithmic. (I confess to having only skimmed his arguments in Shadows of the Mind, despite owning a beautiful signed first edition of same.)
Second, trick mechanisms such as the one he describes don't make evolutionary sense. (I know, that doesn't count as a strike against it in your book, but he doesn't present his ideas as being in anything but perfect consonance with evolution.) You see, according to Penrose, microtubules possess exactly the physical properties necessary for the apparently non-algorithmic aspects of human thought. But microtubules aren't peculiar to humans, or even to organisms that could remotely be construed as conscious, and yet they have always had that structure. How likely is it that this crucial mechanism has lain fallow for a billion years against the day when a creature could make proper use of it? It has teleology written all over it (I know, I know) but Penrose doesn't see it.
[Aside: Penrose would say that lesser minds do make use of the mechanism for what consciousness they possess. But since their consciousness isn't anything that couldn't be functionally (if not specifically) modelled algorithmically, no quantum hocus pocus was necessary. But since human consciousness does (according to Penrose) require such trickery, the accusation of teleology stands.]
(Although I didn't agree with the conclusions of The Emperor's New Mind, it really is an excellent book on many levels. It lays out the basics of both quantum mechanics and computer science as well as I've seen anywhere.)
The speed of light in classical physics is constant relative to what? To the observer. Or am I mistaken?
Relative to everything in the universe.
Depends on what you mean by algorithmic. Do you consider analog computers to be algorithmic? How about A/D converters? What about the fact that neurons have a quiescent firing rate. Couldn't firing rate be a central component of neural "computing" and wouldn't this make it quiet difficult to model? Perhaps as difficult as it appears to be in reality?
Just as an aside, can you point to any example of a digital computer that has a clock rate of about 100hz, but can run rings around supercomputers in pattern recognition. Analog computers are very fast, but run out of significant digits rather quickly. Doesn't this suggest a corresponding sloppiness in wetware?
I am reading Emperor's New Mind and do not at all come away with this conclusion. I am tempted almost to ask whether we refer to the same book. There is without doubt something deep and mysterious going on at the level of fundamental perception and it is intangible. There must of course be a material aspect to perception and some understanding of this is imperative but it is not appropriate to simply declare that materiality encompasses all because we can't describle or understand something real in non-material terms.
It is a stretch for ordinary humanity, even brilliant ordinary humanity, to try to understand that space-time is an aspect of physicality or that light speed is constant relative to everything else. Neither of these make any sense at all from an everyday perspective, which is the perspective in which we all live. And this is only classical physics. Things become much stranger at the quantum level. Some physicist wag once said that anyone who thinks they understand quantum mechanics hasn't studied it enough. I agree. Einstein was wrong about it. I would go so far as to say that the conscious, everyday perceptional frame of reference and mode of thinking is inadequate to really understand it and that this is because what we experience is not what's really going on at quantum levels.
My position would be that there is some "thing" that is operative, immaterial and intangible at the heart of reality and that anything less than this acknowledgement is denial.