In Maclachlan and Reid's The Arithmetic of Hyperbolic 3-Manifolds, when proving that quaternion algebras are simple, they make use of the fact that $M_2(K)$, where $K$ is an algebraically closed field, is simple.
Could someone point me to a reference or proof of this?
$M_n(k)$ is simple for any field $k$, with no algebraic closure hypothesis. There are various ways to prove this. One is to prove the following more general result: if $R$ is a ring, then the two-sided ideals of $M_n(R)$ are $M_n(I)$ where $I$ is a two-sided ideal of $R$. See this math.SE question for more details.