= Solution
As $C$ is a <unitary matrix>, $C^{-1}=C^\dagger$. Let $M=C^\dagger AC$. Its <conjugate transpose> is $M^\dagger=C^\dagger A^\dagger C=M$, so $M$ and every positive integer power $M^n$ are <Hermitian matrices> and have real <eigenvalues>.
More explicitly, if $Av_j=a_jv_j$ with $a_j$ real, then $C^{-1}v_j$ form an <eigenbasis> and
$$
M^n(C^{-1}v_j)=C^{-1}A^nv_j=a_j^n(C^{-1}v_j).
$$
Therefore
$$
\boxed{\operatorname{spec}\bigl((C^{-1}AC)^n\bigr)=\{a_j^n:j=1,\ldots,n_A\}\subset\mathbb R,}
$$
where $n_A$ denotes the matrix size, to distinguish it from the power $n$.
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