From the point of view of enginieers, the most obvious application of complex numbers is computing alternating currents.

Consider first direct current. If you have a network of resistors, and want to compute the current in this network, or the potential of a node, then Kirchhoff's rules reduce this problem to a system of linear equations. Kirchhoff's rules are obvious, essentially saying that ellectric current cannot just disappear.

If you have alternating current, you have capacities and inductions in addition to the resistors, but if you consider them as imaginary resistance depending on the frequency, the computations are exactly the same as in the direct case, just over another field. The alternative would be computing the phase shift separately from the current, which is much more effort and only works for very simple networks, e.g. oscillators.

Once you learned Fourier analysis, this approach immediately tells you how a filter works, and whether a given network acts as a filter.