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 $\mathbb{C}$?

Starting with the very well known iteration formula that creates the Mandelbrot set
$z_{n+1}=z_n^2+c$,
where $c=z_0=x_0+iy_0=\text{Re}(c)+i\text{Im}(c)$ is a complex constant which is the starting point of the trajectory generated in the complex plane by the iteration, in Doppelpot , from the German blog Fraktale Welten by  Nachtwaechter, it is explained that, when we have two complex exponentiations the fractals generated by the recursive formula are normally beautiful, particularly those of  Julia sets.
I suggested  to the author to try the following formulae:
$z_{n+2}=z_{n+1}^{2}+z_{n}+c$
$z_{n+1}=z_{n}^{z_{n}c}$
$z_{n+1}=z_{n}^{z_{n}+c}$
$z_{n+2}=z_{n+1}^{3}+c^{z_{n}}$
$z_{n+1}=z_{n}^{c}$
In Fünf Formeln the fractals are shown.
I reproduced some with the author's permission in this entry of my blog. 

This one is generated by  $z_{n+2}=z_{n+1}^{2}+z_{n}+c$:

and this by $z_{n+2}=z_{n+1}^{3}+c^{z_{n}}$

