In classical E&M, it seems to me that the only point of using
complex numbers is to simplify notation, e.g. to be able to write
a Fourier analysis with only one function, the exponential, rather
than 2, sines and cosines.
On the other hand, a quantum mechanical electron's
wavefunction happens to obey the same rules as complex
arithmetic. The theory is actually incomplete without
complex numbers, in the same way that you can't make
a complete theory of the quadratic equation without them.
Since the original question was about pedagogy, this seems
to me a good argument for waiting until QM to introduce
complex waves.