I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties).
Regardless, I'm wondering if there might be a 'trick' to create a 3D Mandelbrot which has detail running through each dimension without any discontinuities or 'smeared' sections. In 2008 I wrote a short article discussing the possibility, and went on to help discover the Mandelbulb in 2009.
As mentioned in those articles, variations on the quaternion Julia 4D fractal unfortunately resemble 'whipped cream' and have detail running through only 1 or 2 dimensions. Other attempts at a 3D analogue are mere extrusions or lathes of a 2D Mandelbrot. The real thing (if it exists) would look MUCH more interesting and beautiful.
Even the new 'Mandelbulb' isn't perfect as it too contains 'whipped cream' and has less variety than even the 2D Mandelbrot.
Is a Mandelbrot-looking equivalent in 3D space even remotely possible? Here's an artist's impression (created by Marco Vernaglione):