On Similar Matrices' Transformations

Question

Say we have $A,B,C,D\in GL_n(\mathbb{C})$ such that $A$ and $B$ are conjugates,$C$ and $D$ are conjugates, can we find a matrix $P$ in $GL_n(\mathbb{C})$ such that $A=PBP^{-1}$ and $C=PDP^{-1}$?

Answer 1

No. The desired property implies that $AC$ is similar to $BD$, but this is not a consequence of the similarities between $A$ and $B$ and between $C$ and $D$. E.g. consider $$ A=B=C=\pmatrix{1\\ &0},\ D=\pmatrix{0\\ &1}. $$ Then $A$ is similar to $B$ and $C$ is similar to $D$, but $AC\ne0$ is not similar to $BD=0$.