Complex numbers are actually a very poorly taught section in mathematics ... To really start to understand complex numbers you need a good understanding of what a Group, Ring and Field are; I won't go into details but these are (essentially) ways to define sets of numbers that have certain properties.
Now, the field of complex numbers is equlivalent to the field of 2 dimensional rotational matricies which (essentially) means that for every complex number there is one (and only one) rotation matrix that it can be paired up with, and the properties (addition, subtraction, multiplication and division) are all maintained.
Once again, without going into details, the imaginary number on a complex number is related to the cosine of the angle of rotation and the real number is related to the scaling of the rotational matrix; the odd behavior of the imaginary number is because the cosine function is a periodic function.
