I played with the idea of "complex booleans", but it didn't work very well. The idea was, just as in a complex number you have a + i b, let a "boolean" be a + i b, where a and be are either true or false.

But the problem is, if you set true = 1 and false = 0, and then use multiplication for "and", then (true + i true) and (true + i true) = (1 + i) * (1 + i) = (1 + i + i - 1) = (0 + 2 i), which is not a valid answer. So this "logic" didn't have as many nice algebraic properties as boolean logic.