Imaginary numbers (or complex numbers) have a rather unfortunate name. They're no more imaginary than the numbers you've been dealing with before, in that, they're simply a mental abstraction created to solve certain problems (there is no physical entity that "is" the number one, for example - unless you live on Sesame street). For example, negative numbers arose from a need to solve the equation
x + 1 = 0
Complex numbers arise from the desire to give a meaningful solution to
x^2 + 1 = 0
You can think of them as ordered pairs of real numbers, or points on a plain, with their own set of arithmetic, if it makes you feel any better. They can get a bit hairy, but then so can analysis of the reals. Just stick with it and you'll be fine.