  We give an elementary argument for the well known fact that the endomorphism
algebra $End_Q(A)$ of a simple complex abelian surface $A$ can neither be an
imaginary quadratic field nor a definite quaternion algebra. Another
consequence of our argument is that a two-dimensional complex torus $T$ with
$Q(\sqrt{d})\subseteq End_Q(A)$ where $Q(\sqrt{d})$ is real quadratic, is
algebraic.
