I know this is reeeally late... but better late than never, right? This forum's been kinda quiet lately. I should rile it up a bit.
First, the videos: These corkscrews are not clifford tori, they are actual corkscrew-shaped 2D sheets in 4D. They're a combination of sine/cosine waves and exponential/logarithm functions. The complex exponential/logarithm is periodic, unlike their real analogues.
Second, yes, 4D is required to plot a complex function, because each complex number has two degrees of freedom (the real part and the imaginary part). So to fully represent a function F, you need 2 dimensions to represent the original number, and another 2 dimensions to represent the resulting number after F acts on it. So that's 4D. And that's one of the reasons 4D visualization is so useful: it lets you see complex functions in their full glory. :-) (This does
require being able to visualize 2-manifolds in 4D, though, which I find actually a lot harder than visualizing 3-manifolds because of the way 2D sheets can knot themselves in 4D. Or should I say ways
-- 2-manifolds in 4D can knot in really strange ways... like the Klein bottle and the real projective plane.)