Is unitarity/normality of operator independent of inner product?

Question

There is a problem in my textbook:

Let $A\in\mathcal{L}(\mathbb{C}^2)$ be linear operator with matrix representation in standard basis as $$A = \begin{pmatrix} 2 & 3\\ 4 & 1\end{pmatrix}.$$ Can we find inner product in $\mathbb{C}^2$, such that $A$ will be:

1. Normal operator?

2. Unitary operator?

In case that such inner product exists, find one.

My approach is, since $AA^* \neq A^* A$ and $AA^* \neq I$ such inner product cannot be found. But is it correct?

Edit: The problem is in chapter in my Linear Algebra textbook, which deals with Riesz representation theorem and introduction to adjoint, normal, unitary e.t.c. operators, if it helps.