We have the general linear group $\mathrm{GL}\_{2}(\mathbb{C})$ that acts on the set $\mathrm{M}\_2(\mathbb{C})$ of $(2 \times 2)$-matrices by conjugation.

I want to classify the orbits of this action.

What I know:
$$
 \mathrm{Orb}\_{G}(A) = \{ BAB^{-1} : B \in \mathrm{GL}\_2(\mathbb{C})\} \,.
$$