One can indeed do that, for example have (exotic) probabilities which can be negative, or even greater than one etc..
For a simple introduction check this link
As a bonus, even quantum mechanics can be formulated (at least to some extend) as complex probabilities
Check also this post (which covers a different aspect), but which can be a possible application
adding a comment i made here, since it answers criticism on complex probability, but also puts the discussion into the correct framework.
The very starting words on Kolmogorov's landmark work on axiomatization of probability theory states (not exact quote): "A probability is a number assigned to an event in an event space..". Now use some other number (and adjust the arithmetic accordingly) and you have a probability theory (maybe exotic but not less real)
In fact something very similar to exotic probabilities already exists in conventional (lets say) probability theory. It is called random variables. It seems people who downvoted this answer do not have a grasp of the extend of the random variables concept. What are random variables? Random Variables are ways to assign arbitrary numbers, arithmetic and measure to probabilistic events. What is exotic probability? (e.g complex probability), exactly the same. As some say same difference.
One can ask here, so if random variables already cover the functionality of exotic probability and is already a part of traditional probability theory, why should we be concerned with exotic places? Nice question, it is not necesary, in the same sense it is not necesary to use complex numbers instead just use 2 real numbers with specific rules of combination. Or use an array of numbers with special rules of combination instead of matrices, and so on and so forth. One can indeed do that, they are isomoprhic representations. It just happens some are natural representations and provide intuition, flexibility (and potentialy greater understanding) while others are not.