Four dimensional chess?

My two-year Y!A anniversary, celebrated again with a new question:

For a few months now I've been developing a chess variant which involves four spatial dimensions. I've finally gotten around to typing up a few notes on it and making a couple of diagrams if you're interested:

http://my.ilstu.edu/~bmreini/Chess-basicmath.pdf

My main question is, does the idea of 4D chess interest you? If so, then please have a read through the pdf. I've left quite a bit out really, particularly toward the end, so if you notice something interesting or have a comment, make a note of it here. [That's what my best answer criteria is for now.]

You'll notice I don't pick a solution for the slippery king; which one do you like? Do you have another suggestion? My main objective over the next few months is to decide on one central ruleset, which will be dubbed "TessChess" -- as in tesseract chess; "chesseract" is already taken by another supposedly 4D game -- and any other notable rules will be variants thereof in my mind.

Thanks!

[I've asked this in both the Mathematics section and the Board Games section, to get an impression on whether this is more a chess-players game or a mathematician's one.]