Spherical Coordinates are 3D. You need only two values to get to any point on the surface. But if you want to have a not-fixed radius (which is the case, here), you have your third dimension
Maybe, try to rotate in 4D-style? That might give some nice results...
Though, if you go for 4D, maybe you find a way to unify julia with Mandelbrot (not in the Buddhagram way^^) - that probably will lead to more interesting details, as no artifical extension way has to be done then...
Also, omitting one axis probably might not be the best way... what's about a full 4D -> 2D projection? (Or is that equivalent to dropping an axis?)
As said before, doing that direct conversion from 2D to 3D seems a bit odd...
The Mset lies inside an algebratic vector-space, where new calculation rules apply, which results in that nice shape.
Adding just another axis which does the very same as the second axis, where no correspondance is between the two axes, whill likely destroy details. (Especially on the M²-Set, as that doesn't even have radial symmetries)
In a true extension, all axes would comunicate with each other, I guess...
Quaternionic has the "problem" of 2*4D, rather than the 2*2D in Complex numbers (2* because there are the two real and the two imaginary parts)
Maybe, we could try to develop a small algebra just for that...
that would be the simplest case I can think of.... (a*b*c=0 to avoid a fourth Dimension being formed, a, b and c are axial units like i,j and k with the quaternions)
Of course, it's very likely, that this algebra wouldn't hold under any conditions, but for a Mandelbrot-Set, it should be possible to work with
x,y,z are the units from above...
-j²z + 2*j*k*x + 2*j*n*y -k²y + 2*k*n*z -n²x + a*x + b*y + c*z
split into the three parts:
nx -> (-n²+2jk+a)x
ky -> (-k²+2jn+b)y
jz -> (-k²+2kn+c)z
not actually toooo hard formulae... they seem to be closely related to the imaginary part of the Mandelbrot, though. If I'm unlucky, this will nearly look like the quaternion variant...
As long as no exponential or stuff is needed, which you'd first have to figure out, this algebra should hold easily...
for instance, x*x*y*z*z*z=x²*y*z³=-x*y*-z*z=-x*y*-(z²)=-x*y*z=0
x | n is an odd
-x | n is an even
scalars in front of the units simply get extended with the very same rules as always...