Normal Operator that is not Self-Adjoint
Solution 1:
If you're taking $\;\Bbb R^2\;$ wrt the usual, Euclidean inner product, then the "standard" basis is an orthonormal one there and wrt it it's very easy to find the adjoint:
$$T=\begin{pmatrix}2&\!-3\\3&\;2\end{pmatrix}\;\implies T^*=T^t=\begin{pmatrix}\;2&3\\\!-3&2\end{pmatrix}$$
And now just check that indeed $\;TT^*=T^*T\;$ ...