We know that any compact subgroup $G$ of $GL \left(2, \mathbb{C} \right)$ is conjugate to a closed subgroup of the $U(2)$ group. Since our group G is normal, it is simply a closed subgroup of the $U(2)$ group. So we know, that:
$$ G \subset U(2) \subset GL(2, \mathbb{C}) $$
Is it possible to somehow describe all connected $G$, which are a normal subgroup for $GL(2, \mathbb{C})$?