Consider a set $M$ of all possible square matrices of dimension $k$ over a finite field $F_p$. Consider a map $f$ defined on $M$ as $f(X)=X^2+C$ where $X \in M$ and C is an arbitrary fixed matrix from the set $M$.
It is worth mentioning that the operations addition and multiplication on M are over finite field $F_p$.
How to determine analytically the fixed points and periodic points of different period.
Answers even in case of $p=2$ is highly appreciated.
Just to get you started. $k=p=2$
$C=\left(\begin{array}{cc}1 & 0\\ 0 & 0\end{array}\right),\left(\begin{array}{cc}0 & 0\\ 0 & 1\end{array}\right),\left(\begin{array}{cc}1 & 1\\ 0 & 0\end{array}\right),\left(\begin{array}{cc}1 & 0\\ 1 & 0\end{array}\right),\left(\begin{array}{cc}0 & 1\\ 0 & 1\end{array}\right),\left(\begin{array}{cc}0 & 0\\ 1 & 1\end{array}\right),\left(\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right),\left(\begin{array}{cc}1 & 1\\ 1 & 0\end{array}\right),\left(\begin{array}{cc}1 & 0\\ 1 & 1\end{array}\right),\left(\begin{array}{cc}1 & 1\\ 0 & 1\end{array}\right),\left(\begin{array}{cc}0 & 1\\ 1 & 1\end{array}\right)$
no solutions.
$C=\left(\begin{array}{cc}0 & 0\\ 0 & 0\end{array}\right)$, 8 solutions: $X=C,\left(\begin{array}{cc}0 & 0\\ 0 & 1\end{array}\right),\left(\begin{array}{cc}1 & 0\\ 0 & 0\end{array}\right),I,\left(\begin{array}{cc}1 & 1\\ 0 & 0\end{array}\right),\left(\begin{array}{cc}1 & 0\\ 1 & 0\end{array}\right),\left(\begin{array}{cc}0 & 1\\ 0 & 1\end{array}\right),\left(\begin{array}{cc}0 & 0\\ 1 & 1\end{array}\right)$.
$C=\left(\begin{array}{cc}0 & 1\\ 0 & 0\end{array}\right)$ or $\left(\begin{array}{cc}0 & 0\\ 1 & 0\end{array}\right)$, 2 solutions: $X=C,C+I$
$C=I$, 2 solutions: $X=\left(\begin{array}{cc}1 & 1\\ 1 & 0\end{array}\right),\left(\begin{array}{cc}0 & 1\\ 1 & 1\end{array}\right)$
$C=\left(\begin{array}{cc}1 & 1\\ 1 & 1\end{array}\right)$, 2 solutions: $X=C,\left(\begin{array}{cc}0 & 1\\ 1 & 0\end{array}\right)$.
I hope I have understood you correctly. The above does not look promising for a neat general solution!