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Maybe something related to this: http://en.wikipedia.org/wiki/Quasiconformal_mapping, but I don't know anything about it.

>> doubling a rotation to me is pretty much equivalent to rotation (actually it's equal. Only the amount of rotation changes, depending on where the point is found)
doubling a rotation stretches out any shape, just rotating the whole coordinates doesn't stretch anything.
This is what is so special about the 2d mandelbrot, the Z^2 doesn't stretch any shape (that is small enough) no matter where it is in the complex plane.

BradC, quasi-conformal maps sound like they might be way to go. It seems to mean that stretching is allowed but it is finite / bounded.

So the set of operations listed above seems unlikely to be able to produce interesting fractals as it doesn't have two attractors, everything seems to converge on one point (a single branch point I guess).
The other option is to find the *most* conformal 3d fractal, such as http://www.fractalforums.com/3d-fractal-generation/a-new-3d-mandelbrot-like-fractal/?action=post;quote=9827;num_replies=9;sesc=d3a594eba608ec813a9dad70a2e9fdf2
which is probably about '2 quasi conformal' meaning small spheres map to ellipsoids with eccentricity (long radius / short radius) no greater than 2.
Does anyone know of any 3d mandelbrot-like fractal with conformality < 2 ? (1 is fully conformal like 2d mandelbrot, mandelbulb is infinity-quasi conformal).
Even so, the above link shows that it needs to be more than just nearly conformal, the mapping probably needs to be smooth (infinitely differentiable)... and I'm not sure that it is even possible to have a continuous 3d mapping with conformality < 2.

So what other options? Perhaps in 3d one shouldn't look for it being conformal in euclidean space, but conformal in some other space.

there, you quoted it!

aaaaaaaaaaargh