Could quantum mechanics work without the Born rule? 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.
 A: There is a paper called Ruling Out Multi-Order Interference in Quantum Mechanics that, I think, answers your question in the negative (within a certain bound anyway). The authors show that the Born rule implies quantum interference comes only in pairs of possibilities (second order interference), and that by relaxing the Born rule one would expect higher order interference terms in probability calculations.
The authors conduct a three-slit photon experiment and find that the magnitude of the third order interference is less than $10^{-2}$ of the expected second order interference. 
A: The irreducible empirical core of quantum mechanics is a probability calculus. It correlates the outcomes of measurements, so that one measurement (usually called the system's preparation) can be used to calculate the probabilities of the possible outcomes of another measurement. At the center of this probability calculus is the Trace Rule (a special case of which is the Born Rule). If you take aware this Rule, you reduce quantum mechanics to pure fiction, since you have lost your only link between the mathematical formalism and what happens in the actual world.
A: Scott Aaronson (researcher and well-known blogger on topics related to quantum computing) has some lecture notes where he discusses this in a kinda conversational format. Here's the link to the relevant lecture : http://www.scottaaronson.com/democritus/lec9.html
