It struck me, when pondering what it would mean for something like a LLM to have an identity, that this identity as a single path notion will have to be replaced with identity as something capable of branching of into different paths.

Some interpretations of quantum mechanics also points towards such a notion of identity, even if just a philosophical notion

In a sense most of us already identify as a composite of past selves all embedded into our current self, preparing for our future selves. Why is another copy, or branch, going to cause an identity crisis?