Do Christians and Muslims worship the same God?

I don't really care to debate theology, but I thought this was an interesting question when I realized the answer actually depends on whether or not you believe either religion, and it's opposite who I usually hear each answer from.

Assuming we agree that the two religions share a common origin, then if you don't believe in the Christian or Muslim god (and let's include Judaism too), then you can look at the history and development of the stories and, like with evolution and speciation, determine if/where the descriptions of God diverged enough to consider them unique, separate characters.

However, if you believe in one of the religions in question, then it seems that if you recognize the common origin and shared stories (and don't believe the potentially polytheistic origins of early Judaism), then the most reasonable conclusion is that all those other religions believe in the same god as you, but misunderstand/interpret him differently.

So, short answer, if you believe in this god, then yes, Christianity, Islam, and Judaism all worship the same god, but if you don't believe in this god, then potentially no, the stories may have diverged enough to consider them different figures.