Slightly inspired by this question about the historical origins of the Born rule, I wondered whether quantum mechanics could still work without the Born rule. I realize it's one of the most fundamental concepts in QM as we understand it (in the Copenhagen interpretation) and I know why it was adopted as a calculated and extremely successful guess, really. That's not what my question is about.
I do suspect my question is probably part of an entire field of active research, despite the fact that the theory seems to work just fine as it is. So have there been any (perhaps even seemingly promising) results with other interpretations/calculations of probability in QM? And if so, where and why do they fail? I've gained some insight on the Wikipages of the probability amplitude and the Born rule itself, but there is no mention of other possibilities that may have been explored.