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A while ago in the thread on the 3D Mandelbrot I mentioned that the rotational symmetry around the real axis for quaternions could be removed by modifying the axes in some way on each iteration e.g. swapping them or negating them.

Here's one of the simplest forms of this, negating z in (x,y,z,w) prior to plugging the value into the standard quaternionic Mandelbrot q^2+c:

http://www.deviantart.com/deviation/44454355/