If $A$ is a countable dimensional $\mathbb C$-algebra, then the endomorphism algebra of a simple $A$-module is isomorphic to $\mathbb C$. But $\mathbb C(t)$ is an uncountable dimension $\mathbb C$-algebra and since it is a field, its regular module is simple. But $\mathrm{End}_{\mathbb C(t)}(\mathbb C(t))\cong \mathbb C(t)$.

The real issue here is that there are no countable dimension division algebras over $\mathbb C$ besides itself.