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Say I have a small unit of pink horrors that die in one of my phases, either to dangerous terrain or overwatch, which causes them to immediately split rather than at the end of the phase. Can the new unit move / charge in that same phase?

From the wording, RAW, there is no answer to your question. As it is not limited in it's text like say Tyranid Tervigon spawn or deal on summons arrival via deep strike, I would say they can act normally.

I cant find anything* in the rule book as a general rule. That is interesting that they didn't specify "deploys like, as" etc.

*But that doesn't mean it isn't there.

Cheers, that was my reading of it too. I just wanted to make sure that I hadn't missed anything.

On the one hand it doesn't say you can't use them as normal, on the other hand they usually put something along the lines of "May act as normal..." if they are allowed to act as normal. Might be an oversight by those writing the rules, making RAI they can't, but RAW they can.