the short answer is "not that i know of", but that has a lot to do with the nature of 3d vs 2d rendering...
one of the biggest differences between 3d and 2d fractals is that you no longer derive most of the beauty from colouring, but instead from structure, depending on how you render it.
for example, the recent apophysis 3d hack leverages the huge variety of fractals associated with flame rendering and is reasonably flexible wrt colouring, however what you have in the end is nebulous clouds (unless you're really careful with your viewing angle etc). here's an example rendered with chaoscope: http://silent--dreamer.deviantart.com/art/Plasmic-Tornado-2-15520497
on the other hand, when you render solid 3d fractals, you get a real impression of depth with all the usual spatial cues (shadows, occlusion, ...), but you're more restricted in the sort of colouring you can do (usually) and even minor zooming can pose major technical difficulties. here's a recent example from a friend of mine: http://aexion.deviantart.com/art/Furnace-64157856
the video you linked was really interesting btw, and highlights some of these points well:
1. much research has been invested into creating that fractal!
2. it's a "once off" thing: it's a single, particular fractal, not a broad family. changing the colouring doesn't really change this (c.f. 2d orbit trap colourings).
3. the shading is a pretty typical early computer graphics model (diffuse / phong).
4. zooming is very limited (you eventually see spheres), and doesn't reveal much variety.
i suppose a lot of people are looking for the "holy grail" of 3d fractal rendering:
1. huge family of varied fractals, with intuitive controls for exploration.
2. flexible colouring / shading.
3. fast and realistic rendering, especially for zooms.
there aren't any renderers i'm aware of that can meet these brutal criteria, let alone fractal types! in summary: at the moment making 3d fractals requires quite a lot of specialist knowledge for a once-off result, unless you're willing to temper your ambitions: chaoscope, xenodream. this "fixed menu" has been around for some time (at least the 3d apophysis hack will inject some variety!), and all of these renderers have their shortcomings.
finally, i hope to attack this problem properly after my exams
i've got a pretty high-powered rendering system framework in place (effectively solving point 3 on the "holy grail" list), and have ideas for 1 and 2...
hmm that's a lot more than i planned to write
can you tell i'm trying to dodge studying?
 the structure vs colouring topic was recently brought up in this thread: http://www.fractalforums.com/index.php?topic=738
 by this i mean ifs rendering la scott draves: http://flam3.com/flame.pdf