$A$ is similar to $\begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$

$A$ is similar to $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$

$A$ is similar to $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$

$A$ cannot be diagonalized over $\mathbb{C}$