Use of complex numbers in ee generally comes down to representing sinusoids by an amplitude and phase, abstracting away the high frequency carrier. So, we call a cos(w t+p) as just (a angle p). We can momentarily think of (a angle p) as a vector. Its rectangular form is (a cos(p), a sin(p)).
Operations of addition and subtraction are defined for vectors and have physical meaning. (Remember adding and subtracting must be done in rectangular, not polar coordinates).
For example, adding two waveforms a cos(w t) and b sin(w t) would be represented as (a angle 0) + (b angle -90), or (a, 0) + (0, -b) = (a, -b). We can then convert back to polar as (sqrt(a2 +b2 ) angle atan(-b/a)). We can then go back to the original form, sqrt(a2 +b2 ) cos(wt + atan(-b/a)), which should be equal to the original proposed sum unless I've mucked something up.
Thinking of the vector (a angle p) as a complex quantity allows division of these vectors in the expected way. You could also think of just defining the correct rules for dividing the vectors, bypassing complex numbers altogether (some people teach it this way), but it turns out they're pretty damn useful, and why reinvent the wheel?
To get the complex form, just plop the vector down onto the complex plane. For example, (a angle p) = (a cos(p), a sin(p)) becomes a cos(p) + 1i a sin(p).
We can go through the same exercise to show that division of two sinusoids can be done on the complex number form and then translated back to full form, giving the correct answer. But, this is getting a bit long, so ask if you have trouble with it.
We have to be a bit careful with multiplication of these vectors. The vector form kind of breaks down if you multiply time dependent sinusoids because different frequencies emerge from the operation. Multiplication of a sinusoid by a ratio of same-frequency sinusoids is permissible. Impedance is defined as the ratio of voltage to current---time dependence drops out and we are left with a complex number.
The point of all this? It saves a lot of time, has cleaner notation, makes it easier to think about complex systems, and makes solving ODEs often trivial (for that you need Euler's identity). In linear systems, nothing can change the carrier frequency of a signal, so the complex representation is all you really need.
I'm probably glossing over a few things and welcome feedback and criticism.