By matrix multiplication , $A^2$ = $\begin{bmatrix} 3 &2i \\ 2i & -1 \end{bmatrix}$ whose trace is 2.
$A^3$ = $\begin{bmatrix} 4 & 3i\\ 3i & -2 \end{bmatrix}$ whose trace is 2.
So we see the trace of the original matrix remains the same irrespective of repeated multiplication with itself. Therefore trace of $A^{10}$ = 2