Consider the following two equations
$X = M + \eta_1$
$Y = M + \eta_2$
where, $X\in\mathrm{R}^{n\times n}$, ia a real random matrix with mean $M\in\mathrm{R}^{n\times n}$. $\eta_1$ is Gaussian white noise with mean $0$ and covariance $\sigma^2I_{n\times n}$. while $Y\in\mathrm{C}^{n\times n}$, ia a complex random matrix with the same mean $M$. $\eta_2$ is complex Gaussian white noise with same mean $0$ and covariance $\sigma^2I_{n\times n}$. As such, $X$ and $Y$ have same mean but they differ in the nature of noise. Will the eigenvalues of the two matrix $X$ and $Y$ have same mean or will they differ?
The sum of the eigenvalues (counted by multiplicity) is the trace, and that is linear, so the means of the sums of the eigenvalues of $X$ and $Y$ are both equal to the sum of the eigenvalues of $M$. Apart from that, I don't understand the question. Each matrix has $n$ eigenvalues, usually distinct and complex. There is no canonical way to single out a particular eigenvalue of $X$ or $Y$, so how can you ask whether they have the same mean?