This looks an awful lot like a 3D version of the tricorn to me.

http://en.wikipedia.org/wiki/Tricorn_%28mathematics%29

It's made with the new function using a phase of 0, p_r=2, p_theta=-2, 
p_phi=2 at ten iterations.  (Note the negative p_theta.)

On another note, Paolo Bonzini recently published a paper describing how 
to express the mandelbulb function variations as a quaternion function 
mapping.  I generalized his expression to work with any scalar power, 
and besides being more flexible, it looks like it will be more 
computationally efficient than the non-integral exponent versions of the 
White, Nylander, et. al., formulae (the ones currently in use).

This quaternion version requires only 1 atan2, 1 cos, 1 sin, 2 sqrt, 1 
fractional pow(), and 3 scalar divisions per iteration (along with a 
bunch of fast multiplies and adds).  Unfortunately, the expression 
becomes extremely cumbersome when attempting to shoehorn it into a 
single SDL function, so no working code yet.

I also plan to flirt with insanity by taking a crack at reworking it for 
full quaternion power exponents.

~David

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