Always wondered what Skynet would have done had it killed off the humans in the Terminator future. Would it build a machine society?

There are some acceelerationists who want "superior" machines to replace humanity. But what if instead they get the equivalent of a paper clipping gray goo, or a bored Skynet who just fights it's own creations? Is that worth replacing all the humans?

As a thought experiment, what is a p doom AI doing after humanity is gone?

There are many insect/bird/mammal species that exhibit reliably predictable behaviors. It will co-opt them.