  Let $G$ be a Kac-Moody group over a finite field corresponding to a
generalized Cartan matrix $A$, as constructed by Tits. It is known that $G$
admits the structure of a BN-pair, and acts on its corresponding building. We
study the complete Kac-Moody group $\hat{G}$ which is defined to be the closure
of $G$ in the automorphism group of its building. Our main goal is to determine
when complete Kac-Moody groups are abstractly simple, that is have no proper
non-trivial normal subgroups. Abstract simplicity of $\hat{G}$ was previously
known to hold when A is of affine type. We extend this result to many
indefinite cases, including all hyperbolic generalized Cartan matrices $A$ of
rank at least four. Our proof uses Tits' simplicity theorem for groups with a
BN-pair and methods from the theory of pro-$p$ groups.
