One way to read Schur's Lemma is
The endomorphism ring of an irreducible representation of a finite group over a field $K$ is a division algebra over $K$.
For $K=\Bbb R$ there are easy examples where the endomorphism ring is isomorphic to either $\Bbb R$, $\Bbb C$ or $\Bbb H$ (the quaternions). For $K=\Bbb C$ there is not much to check as there are no non-trivial finite-dimensional division algebras.
What is the general situation? Can every division agebra be realized as an endomorphism ring of some representation?
Qustion: Given a finite-dimensional division algebra $\mathfrak A$ over $K$, is there a representation of a finite group over $K$ whose endomorphism ring is isomorphic to $\mathfrak A$?.