Is there such thing as an imaginary (imaginary number)? In other words... is there such a thing that is to imaginary numbers what imaginary numbers are to real numbers? And could this be expressed as a "complex" type number? If a complex number is in the form x + yi, I guess this would be in the form of x + yi + zj?
Does that exist as a concept?

In a sense, yes. There is a number system called the Quaternions, which is an extension of the complex numbers. You can find the Wikipedia page here: http://en.wikipedia.org/wiki/Quaternion
In the complex numbers, we have $i^2 = -1$, but in the quaternions, which extend to $i, j, k$, we still have $i^2 = -1$, but we also have different rules such as $ij = k$ and $jk = i$. This number system plays a large role in simplifying engineering equations, just as complex numbers simplify equations describing electromagnetism and fluid dynamics. 
Certainly, you can extend this system to arbitrarily many variables, so long as the basic rules (axioms) are consistent. And really, that's something mathematicians do: Create structures, give it basic rules, and see what comes out of that.
