Can the same vector be an eigenvector of both $A$ and $A^T$?
Let $n\geq 2$ and $Z_n=\{A\in M_n(\mathbb{C}); A,A^T \text{have at least one common eigenvector}\}$.
Proposition. $M_n(\mathbb{C})\setminus Z_n$ is a Zariski open dense subset. That implies (for example) that if you randomly choose the $(a_{j,k})=(\alpha_j+i\beta_k)$ according to a normal law, then $A,A^T$ have no common eigenvector, with probability $1$.
Proof. According to Shemesh, cf. Remark 3.1:
$A\in Z_n$ IFF $(*)$ the matrix $[[A,A^T]^T,\cdots,[A^k,{A^l}^T]^T,\cdots,[A^{n-1},{A^{n-1}}^T]^T]$ has rank $<n$. Since $(*)$ can be written as a complex algebraic system of relations, $Z_n$ is a Zariski closed subset. It remains to show that, for every $n$, $Z_n$ is not $M_n(\mathbb{C})$.
Choose $a_{j,j}=j$ and if $j<k$ then $a_{j,k}=1$, the other $a_{i,j}$ being $0$.