I'm reading about the Sato Tate conjecture for genus 2, and I came to the paper here. This breaks the conjecture into 6 different parts, based on the real endomorphism algebra of the surface, or the connected component of the Sato Tate group. They point to the classification here, which claims in section 4.1 that "Albert’s classification of division algebras with involution (see [Mu70]), together with the work of Shimura [Sh63], show that the R-algebra $End_{\mathbb{R}}(A_K)$ is isomorphic to one of" $\mathbb{R},\mathbb{R}\times\mathbb{R},\mathbb{R}\times\mathbb{C},\mathbb{C}\times\mathbb{C},M_2(\mathbb{R})$ or $M_2(\mathbb{C})$.
Conspicuously not on the list is $\mathbb{C}$. Is there a good reason for $\mathbb{C}$ to be able to be the endomorphism algebra of an elliptic curve, yet not an abelian surface? Why can an abelian surface have real multiplication but not complex multiplication?
