Well, for one thing, for such quantity to make physical sense, both the real part and the imaginary part should be of the same dimension, e.g. "length." Also, the result of a measurement is supposed to come from (be an eigenvalue of) an observable - an operator, and, on the one hand, I think I'd have a hard time conjuring one up; on the other hand, the eigenvalues are "supposed to be" real anyway! So, no, that doesn't work.