Here's a thing I literally dreamed of last night :D
Let $B=\mathcal{B}(\ell^2(\mathbb{N}))$, or for that matter some other unital C*-algebra. Let $M_B\subset B$ be the set of operators $c\in B$ for which the sequence defined by $a_0=0,\ a_{n+1}=a_n^2+c$ is bounded.
What can one say about $M_B$? Has there been research about it? Is it a frightening looking infinite dimensional analogue of the classical Mandelbrot set? I mean, what certainly is true is that for any projection $p$ and for any element $\lambda$ of the classical mandelbrot set, $\lambda p$ must be in $M_B$, hence $M_B$ contains infiniteley many copies of $M_\mathbb{C}$, but what can one say about operators not of this form?
Thank you in advance!