I have been fooling around with Mandelbrot fractal to and fro for many years. One of the latest years I learned some general algebra with some of the most basic groups, like cyclic groups, dihedral et.c..
I know a little about how complex multiplication on the unit circle is related to cyclic groups and complex multiplication is of course a central part of the Mandelbrot fractal.
Not on my lab computer right now but I will show some of my findings as own work in this question later.
However my question is if there have been formally investigated deeper connections to the behaviour of the Mandelbrot set and modern algebra (or maybe if you have found any yourself)?
In fact, the very first paper with a crude image of the Mandelbrot set was published in 1978 by Brooks and Mateleski and was quite algebraic in nature. They were interested in discrete subgroups of the projective linear group $\text{PSL}(2,\mathbb C)$. (This is simply the group of invertible Mobius transformations under composition, as indicated on this Wikipedia page.)
More specifically, they were interested in when a subgroup of $\text{PSL}(2,\mathbb C)$ generated by two elements might be a discrete subgroup and showed that it boiled down to whether a certain set was a discrete set of $\mathbb C$. That certain set was exactly the limit points generated by the recursive formula $z_{i+1}=z_i^2+C$. Thus, they were led to consider the set of $C$ values where that recursion leads to a stable periodic orbit. You can find a copy of their paper here and near the end is this fabulous picture: