I would think that quantum effects would cause the states of two universes that began in identical states to diverge. At the quantum level particles will flit in and out of existence, but not identically in the two universes. Eventually and inevitably these quantum differences will manifest themselves at the macro observable level.
I think the existence of the quantum world guarantees that true randomness exists in the universe. The Casimir effect is one evidence of the random quantum world, where particles flit randomly in and out of existence, governed only by conservation laws regarding charge and spin and so forth.
That quantum randomness can have a macro impact is obvious - it was the whole point of Schrödinger's cat, and its existence is responsible for a great deal of human activity, most obviously quantum research and papers, and a great many Internet pages. Quantum effects have practical applications, such as tunnel diodes, and so does quantum randomness, such as generating secure cryptographic keys.
I do believe we live in a clockwork universe, but only down to the quantum level where elements of reality begin to become truly random. Quantum effects can affect dice, photons, anything.
What does this mean for determinism? I think it means that what we think and do is largely but not completely deterministic, and that any sense of control we have is an illusion.
Can you though? All your actions could be (and evidence indicates are) completely determined by the initial conditions of the particles in your brain.
In an indeterministic universe, aren't final outcomes only incompletely specified by initial conditions, whether a human or an electron sent toward two slits?
AbE: Just saw Son Goku's Message 108. I believe I just made the same point. Quantum effects easily influence the macro level, obvious examples begin Schrödinger's cat and an experimenter writing up the results of a two slit experiment. In those cases we know the macro level outcomes of quantum influences. The human mind must be continuously influenced by those same quantum influences, but we just have no data telling us the specific quantum events and macro outcomes.