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I'll start with Tglad's mandelbox, generalised to 4 dimensions via a particularly hackish edit to mandelbulber. This is just a basic render from a fixed point in 3-space but moving in the 4th dimension. Later: I plan to do a tidier job of adding the 4th dimension, then add at least one more and do a 5-dimensional fly-through.
<a href="http://www.youtube.com/v/CTD1HNgehYI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/CTD1HNgehYI&rel=1&fs=1&hd=1</a>