16.58 (2006)

No, it is not. It is known that $\mathbb{C}$ has infinitely many (discrete) automorphisms. For $\varphi \in \text{Aut}(\mathbb{C})$ and a matrix $x$, let $x^\varphi$ denote the matrix obtained from $x$ by applying $\varphi$ to each element. Then $T_\varphi : x \to x^\varphi$ is an irreducible representation of $SU_2(\mathbb{C})$. It is easy to show that among these representations there are infinitely many pairwise non-equivalent ones. Therefore the group $SU_2(\mathbb{C})$ itself does not satisfy the condition of the problem. (This observation belongs to von Neumann.) One can show that any group satisfying the condition of the problem is isomorphic to a subgroup of $SL_2(\mathbb{Q})$ satisfying the condition of Problem 15.57. On the other hand, E. Cartan proved that a unitary representation of a simple compact Lie group is always continuous. Hence, if we restrict ourselves to the unitary representations, then $SU_2(\mathbb{C})$ does satisfy the condition of the problem. (V. P. Burichenko, Letter of 16 July 2013.)