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Finding the critical points using f'(z) = 0 for the first formula results in a critical point at zero and the other 10 critical points are dependent on the position in the complex plane. However, the critical point at zero is useless because after the first iteration all subsequent iterations result in 1.

I've checked, those pictures use c as the initial value of z so result in 'perturbed' fractals as c (the location in the complex plane) is not a critical point.

I've added a variation of these formulae to Saturn.

z = (z^alpha + beta)/(z^gamma + delta) + epsilon + zeta

with the following default values:

alpha = 10
beta = position in the complex plane
gamma = 8
delta = position in the complex plane
epsilon = position in the complex plane
zeta = -2
z0 = 0

Pictures soon ...

At some point formulae of this type will be added to Neptune (my multiple critical point fractal program), I have to implement critical points dependent on the location in the complex plane. The critical points will have to be evaluated for all locations instead of once.

Thank you element90 for the explanations and of course the wonderful pictures !

BTW, my previous formula for the iterated derivative f'(c) was erroneous. It has been corrected.