Mandelbrot-like sets for functions other than $f(z)=z^2+c$? Are there any well-studied analogs to the Mandelbrot set using functions other than $f(z)= z^2+c$ in $\mathbb{C}$?


Here's the Mandelbrot set on the Poincaré Disk. I made it by replacing all the usual operations in the iteration
$$z_{n+1} = z_n^2+c$$
by "hyperbolic" equivalents. Adding a constant was interpreted as translating in the plane, and the hyperbolic equivalent is then
$$z \mapsto \frac{z+c}{\bar{c}z+1}$$
For the squaring operation, that meant I used angle doubling plus rescaling of the distance by a factor two based on the distance formula for the Poincaré Disk:
$$d(z_1,z_2)=\tanh^{-1}\left|\frac{z_1-z_2}{1-z_1\bar{z_2}}\right|$$
