The Many-Worlds interpretation, proposed by Everett in 1957, hypothesises that there is no collapse of the wavefunction at the moment of observation. Brief details of the theory are given in the Lazy Layman's Guide to Quantum Physics. The following links are to discussions of the detail of how an MWI 'multiverse' would work:

Causation, and the relationship between universes

Discussion of the concept of self, measure, and rational behaviour, under MWI

What is the MWI and where does it stick its neck out?

What are the moral implications of MWI, if any? *

How does MWI compare with Copenhagen and Time Reversibility? *

The Weak Anthropic Principle *

If we start from the corrected Cogito Ergo Sum - i.e. something like