Given a Hopf algebra over $\mathbb{C}$, and a irreducible comodule $V$, I can show that Mor$(V,V) \simeq \mathbb{C}$, where Mor$(V,V)$ is the space of comodule maps from $V$ to $V$. My proof uses that $\mathbb C$ is algebraic closed.
Does this work over non-algebraically closed fields, i.e. $\mathbb R$?
If you consider the Hopf algebra $k^G$ of $k$-valued functions on a finite group $G$, then $k^G\text{-coMod}$ is equivalent to $G\text{-Rep}_k$, so your question reduces to finding irreducible finite group representations whose endomorphism division algebras are strictly bigger than the scalar field. For this, e.g. see About the Schur's lemma