If i recall correctly you can represent a harmonic oscillator as a linear differential equation with a 2x2 matrix. The imaginary part of the eigenvalues of this matrix correspond to the angular frequency of the oscillator.

I like this example because it gives a physical meaning to both eigenvalues and imaginary numbers. It also shows the connection between the sine and cosine and the complex powers of e comes from (since you can show that all three solve the differential equation).