Wikipedia describes quantum immortality or suicide as the following:
In quantum mechanics, quantum suicide is a thought experiment,
originally published independently by Hans Moravec in 1987 and
Bruno Marchal in 1988 and independently developed further by Max
Tegmark in 1998. It attempts to distinguish between the Copenhagen
interpretation of quantum mechanics and the Everett many-worlds
interpretation by means of a variation of the Schrödinger's cat
thought experiment, from the cat's point of view. Quantum immortality
refers to the subjective experience of surviving quantum suicide
regardless of the odds.
Max Tegmark claimed this was the only way he knew (page 5) to test the many worlds interpretation although the researcher taking the place of the cat would not be able to tell anyone the result since the living researcher would be in a superposition that did not experience the gun firing.
The Copenhagen interpretation, or more generally some single-world interpretation rather than many worlds, would offer panpsychism the most support, because a single-world interpretation would require the wave function to, at least metaphorically, collapse.
If one interprets the wave function collapse as a choice of one alternative from many then one could argue that there is a conscious agent of some sort involved making that choice. From there one might be able to argue for panpsychism. With many worlds the wave function does not collapse and so there would be no reason to introduce the idea of agents making choices.
The idea of a choice suggests free will. John Conway and Simon Kochen in their Free Will Theorem claimed that if we have free will then so does a quantum system. Starting from this paper one might find a way to argue for panpsychism.
See also Shimon Malin's Nature Loves to Hide for an intuitive overview of quantum physics. Malin appears to favor the idea that a wave function collapse involves a choice. He also brings in Whitehead and Plotinus. This might be another way to argue for some kind of panpsychism, but it could go well beyond panpsychism.
It is not clear what is going on in a quantum system. Calling the system a wave when one is not looking at it, but a particle when a measurement has been made seems wrong. Saying there are many worlds to cover up the indeterminacy in the wave model seems even worse. Saying there are agents making choices at the quantum level (panpsychism) seems hard to believe.
Conway, J., & Kochen, S. (2006). The free will theorem. Foundations of Physics, 36(10), 1441-1473.
Malin, S. (2002). Nature loves to hide: Quantum physics and the nature of reality, a Western perspective.
Tegmark, M. (1998). The interpretation of quantum mechanics: Many worlds or many words?. Fortschritte der Physik: Progress of Physics, 46(6‐8), 855-862.
Wikipedia, "Quantum suicide and immortality" https://en.wikipedia.org/wiki/Quantum_suicide_and_immortality