The question whether we could have a logical system that can be represented with complex numbers raise an interesting point: Are the logical systems where multiple dimensions are useful?
The answer turns out to be yes. Consider the multidimensional logic of Carlos Gershenson.
Here, each logical variable is a pair from the 'square' [0,1] x [0,1]. The reason that a 2-dimensional representation is chosen is such that we can assign a truth value to even paradoxical statements such as "This phrase is false." The basic idea is that if for the pair (x,y) we have x+y=1, then this is considered a non-paradoxical value within fuzzy logic1. Otherwise, the truth value is paradoxical, but can still be represented and computed with. (for more information, see the link provided)
But let me answer your actual question. One of the main reasons that most of mathematics uses a two-valued logical system is that most of mathematics is concerned with proving something either true or false. Nothing else. Hence, as mathematicians only wish to speak about two logical values for their statements, a two-valued logical system is the simplest system that allows them to do that.
1: Here we see a parallel with the 'imaginary numbers', they were introduced in Cardano's formula as an 'algebraic trick' to have some 'nonsense' in the middle of a derivation, but a correct result at the end)