Well I have the proof with complex matrices.
I was wondering if it held that since two real matrices that are similar in $M_n(\Bbb C)$ are similar in $M_n(\Bbb R)$ and since $M_n(\Bbb R)$ is closed in $\Bbb M_n(\Bbb C)$, then it follows that $Sim(M)$ is closed iff $M$ is diagonalizable in $M_n(\Bbb R)$, but I don't think that's enough.
(By the way, is diagonalizable the right term ? It seems really weird)