Ever since we've been using exponentials to work with electromagnetic waves, I've been confused about the imaginary portion and want to confirm my thinking.
What does the imaginary portion represent? Nothing, right? It's just a side effect of using complex exponentials because they are very easy to deal with algebraically. So, in reality we can completely restructure all the math to be written in terms of cos/sin instead and never let a single imaginary number appear, right?
The exponential function is easier to manipulate than the real trigonometric functions, in particular when it comes to derivatives and integrals: the manipulations can be done completely algebraically using complex numbers. In practice, it's easier to manipulate $e^{i\alpha}e^{i\beta}=e^{i(\alpha+\beta)}$ than
$$
\cos(\alpha)\cos(\beta)=\frac{1}{2}\left(\cos(\alpha-\beta)+\cos(\alpha+\beta)\right)
$$
etc.
Of course the physical signal is real, which means that one must eventually return to either cosine or sine form. For this, a convention is chosen, and the most common one is to take the real part of the exponentials and ignore the imaginary part. One can then include various effects by using complex propagation constants, complex permittivity etc.
Yes, it is out of convenience and to simplify the equations. The implication is that the real part of the complex quantities is taken to get the actual physical value (which gives you some term that is a cosine with some phase).
This is analogous to the way we encode the phase of our quantities in complex alternate current calculations and similarly the amplitude is given by the absolute value of our complex quantity and the phase is given the argument of the complex number.