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Again, I am slow in getting my thoughts into the aether. Following is what
I wrote yesterday but did not post. I'll then continue to answer your post
in my next, which, with luck, will appear shortly.

Charles, you asked 'why 4D chess?' Because it's beautiful.

I mean that sincerely. Some configurations pieces take during play in [my
version of] Hyperchess are fascinating and can be actually beautiful. Well,
to a CV player, anyhow. 

But you are right that these games don't get much play. There's only one
complete GC game here and it ended when one player let college distract him
from the game, blundered badly, and then resigned. If I recall, every
completed game I know of has ended in resignation, not mate. 

I cannot honestly say that mate is all that achievable, although with the
'Held King' rule it is certainly possible. And that is the key to the
reason most people who try these games don't play many - not only are they
relatively hard to play, but they are almost impossible to end, because a
higher D king is so insanely mobile. What do you do about the 'Slippery
King' problem? How one solves that problem determines a lot about the
game.

What characterizes a satisfactory solution to the problem of checkmate?
I'd like to suggest that it do as little violence to the standard game's
concept of checkmate as possible. By restricting the game to only 3
dimensions, you make it easier to track down the king, somewhat. [And I
have been assuming they are all spatial dimensions. If you are looking for
a time travel game, Gary Gifford has an excellent one.]