The Everett multiverse interpretation of Quantum Mechanics has two parts. First, there is the dynamics: the wavefunction never collapses. Second, there is an interpretation of the dynamics: superpositions (with respect to the privileged basis) can be thought of as multiple worlds, so that the world constantly splits and splits, and we become more and more multilocated. The dynamics solves the problem of the inelegance in the theory induced by collapse, and the oddness of thinking that observers are special vis-a-vis fundamental physics. The interpretation helps solve the conceptual problem of what superposed states mean.
But what if we stick to the dynamics of the Copenhagen interpretation with measurement-induced collapse, but combine it with a multiple worlds interpretation like in Everett? We admittedly do lose the advantages of a no-collapse theory.
However we can still borrow the Everett solution to the conceptual problem of what superposed states mean. Here's how this might work. Let's say that Sally prepares an electron in a mixed positional state, so it's in a superposition of being in box A and of being in box B, with coefficients such that she has a 1/4 probability of finding the electron in A and a 3/4 probability of finding it in B in subsequent experiments. If the preferred basis is positional, then when she prepares that electron, her world branches into two. In one world, the electron is definitely in box A. In another world, the electron is definitely in box B. But unlike in Everett, when she makes the measurement of the positions, one of these worlds ends, in accordance with the probabilities in the wavefunction, and the wavefunction then collapses. So, for a while, Sally was located in two worlds, but then one of the worlds was terminated. However, there never is a superposition of different states of consciousness or of their physical correlates. So all the worlds we're in look the same.
I think this helps with the problem of making biological and geological claims about what things were like before observers true (just indexed to our world). But there are some serious mathematical issues there, so I can't insist on this part.