Let $A$ be a complex matrix and $AA^t=A^2$, is it always right that $A$ is symmetric? It is a well-known result that when $A$ is a real matrix and $AA^t=A^2$ then $A$ is symmetric. I guess the same proposition is right for a complex matrix, but I can't prove it. Could someone give me some hints on this problem? I will be very appreciated for your help!

This is not true in $M_n(\mathbb{C})$, $n\geq 3$.
A very simple counterxample is the following:
Let $v^t=(0,0,1,0,...,0)$ and $w^t=(1,i,0,...,0)$.
Notice that $w^tw=1+i^2=0$ and $w^tv=0$. Define $A=vw^t$.
Now, $A^2=(w^tv)A=0$ and $AA^t=(w^tw)vv^t=0$, but $A=vw^t\neq w^tv=A^t$.
