A few elementary (and probably not very useful) thoughts:

You can always write these simply as polynomial maps of affine space, and then they are special cases of general theorems about polynomial maps of affine spaces. So presumably what you want to know is if you can use the matrix formulation to glean additional information. Have you looked at the $n=2$ case. Writing $X=\left(\begin{smallmatrix} x&y\\ z&w\\ \end{smallmatrix}\right)$
and $C=\left(\begin{smallmatrix} a&b\\ c&d\\ \end{smallmatrix}\right)$, the map $X\mapsto X^2+C$ is simply the map
$$
F : \mathbb{A}^4\to\mathbb{A}^4,\qquad
(x,y,z,w) \mapsto (x^2+yz+a, xy+wy+b, xz+wz+c, w^2+yz+d).
$$
Homogenizing gives $\bar F : \mathbb{P}^4\to\mathbb{P}^4$,
$$
\bar F(x:y:z:w:t) = (x^2+yz+at^2: xy+wy+bt^2: xz+wz+ct^2: w^2+yz+dt^2 : t^2)
$$
The indeterminacy locus on the hyperplane at infinity is the rational curve parametrized by (if I've computed correctly)
$$
\{ (uv, u^2,-v^2, -uv, 0) : u,v\in\mathbb{C} \}.
$$

What do you mean by "the" Mandelbrot set in this setting? I guess one could define it to be the points $C$ such that the orbits of the critical points are all bounded. Of course, the space of $C$ is not 4 dimensional, since you can always conjugate by an arbitrary invertible matrix. If you're working over $\mathbb C$, then you may assume that $C$ is in Jordan normal form, so the moduli space consists of two components, namely diagonal $C$ and $C$ consisting of a single non-semisimple Jordan block. You might consider each of these in turn.