Well that depends on your idea of complex society. The first instance of centralized authority in Mesopotamia shows up as governor-kings who ruled over one or a few cities, and they don't show up until the early dynastic period, or ~3000 b.c.e, which is around the time Egypt is being unified by Narmer(Menes), so Egypt was already pretty civilized. But Mesopotamia did develop a lot of things we associate with civilization before Egypt, such as writing, domestication of animals and wild vegetation and the like, mostly because there was a wider degree of wildlife in early Mesopotamia and that Mesopotamia was easy to access for trade, which is what many people believe led to the development of writing.