Imaginary Number in Logic

There is a vast literature on many-valued logics. Lukasiewicz's original 3-valued logic is perhaps the simplest such logic and the extra truth value $P$ satisfies your formula $P \iff \lnot P$. Lukasiewicz developed this into what are now called Lukasiewicz logics that have been intensively studied and generalised over the years. One of these generalisations is the subject of fuzzy logic, which has practical applications, e.g. to model situations where knowledge is imperfect.

See http://plato.stanford.edu/entries/lukasiewicz/ for more information about Lukasiewicz's work and http://plato.stanford.edu/entries/logic-manyvalued/ for a survey of many-valued logics.

As Rob Arthan says, there's a vast literature on this topic, only some of it in application to the paradoxes.

As a solution to paradoxes like the liar, though, the introduction of a third truth-value meets an immediate difficulty, the so-called "strengthened liar":

This sentence is not true.

You may be underestimating what was involved in the creation of the complex number system. It's not just a matter of introducing a symbol $i$ and declaring that $i^2=-1$. Rather, it's important that much of what people knew about real numbers also applies to complex numbers; specifically, $\mathbb C$ is a field of characteristic zero (though not an ordered field). So people could continue to manipulate equations (though not inequalities) involving complex numbers just as they did with real numbers. It turned out that the properties of real numbers that persist when one passes to complex numbers are enough to give an interesting and useful theory (and later, people found additional properties of $\mathbb C$, like algebraic closure, that make it even more interesting and useful).

Analogously, it would do little good to introduce a new truth value and declare it to be equal to its negation. One would need to show that the new system of truth values retains enough of the properties of the traditional system so that people can continue to reason more or less as they are accustomed to do.