I’m used to working with the Frobenius norm of a matrix $A\in M_{2,2}(\mathbb{C})$ defined as \begin{equation} \left\|A \right\|_F := \sqrt{\operatorname{tr}(AA^\dagger)} \end{equation} which is convenient to work with as it is invariant under unitary transformations.
Can one define a matrix norm similar to this which is invariant under $SL(2,\mathbb{C})$ transformations instead of unitary transformations? In other words can one define a norm such that \begin{equation} \left\| A \right\|= \left\| A S \right\|= \left\|S A \right\| \end{equation} for any $S\in SL(2,\mathbb{C})$?
Such a norm cannot exist. Otherwise, for all invertible matrices $A$ we have $$\|A\| = |\det A|^{1/2} . \| \frac{1}{(\det A)^{1/2}} A \| = |\det A|^{1/2} \| 1\|, $$ because $\frac{1}{(\det A)^{1/2}} A \in $ SL$(2,\mathbb{C})$.
Then, for $\| \cdot \|$ to be a norm, we must have that $A \mapsto |\det A|^{1/2}$ is a norm, which is not the case.