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I'll do it with vector algebra, though it might be simpler with complex numbers:
Vector v = (x,y) on the plane.
Start with one quadrant circle:
 (x - 1)^2 + (y-1)^2 < 1
This gives us all four quadrant circles:
 (x,y) = (y,-x)   // defines 90 degree symettry
Next get all the larger and smaller 'quadrant' circles:
 a = (1-sqrt(0.5)) / (1+sqrt(0.5))
 v = av 

Next get the mobius transform:
 b = sqrt(2) + 1
 v = b^2/(v + (b,0)) - (1,0))

It isn't too surprising that a different parameter will give different results... whiter ones probably correspond to 'less simple attractors' or transforms that line up less well.
Nice picture by the way.
Incidentally, the circles appear to be the same, just smaller radius.

Moving my "hijack" of this thread to here: http://www.fractalforums.com/new-theories-and-research/why-there-isn't-3d-12-conformal-transformation/

If this is a question about my particular octahedral fractal - it will look exactly the same. The fractal is based on the octahedron, and octahedron is North-South symmetric:

Another update:

Birth, Life and Death of a Quantum Fractal

<a href="http://www.youtube.com/v/lRHl27H2UpI&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/lRHl27H2UpI&rel=1&fs=1&hd=1</a>