On the one hand, Dawkins is a rather vocal exponent of evolution. Evolution, to me, appears to hold that greater complexity can arise form lesser complexity (unless we want to argue that the causes conspiring to increase the complexity of a particular part of the universe are summa summarum greater in some sense than the effected increase in complexity in the organism, which isn't obvious to me).
On the other hand, Dawkins requires that God be incredibly complex if he is to be a legitimate cause of the universe, that is, he must be complex enough to produce something like the universe. To him, this contradicts the Christian, or at least Catholic belief, that God is irreducibly simple, composed of no parts, and so on.
Dawkins is not particularly well known for his philosophical erudition, but if we take his claims and evaluate them on their own merit, are there problems between these two positions?