For the wavefunction comment consider that the wavefunction of an electron does not actually vibrate in the "real world." (Most wavefunctions don't.) For that reason alone we need something like a complex number. Spin is another example...One of the spin 1/2 operators needs to be non-real, otherwise the spin measurements make no sense. etc. etc.

Other cases where we can get away with not using an imaginary component but where it is used anyway is typically to turn two equations into a single equation that explains two related things. For waves (optical and otherwise) we can use complex numbers in the index of refraction not only to get intensity but also absorption information. In EE it is common to use the strategy to give information not only about transmission properties but also resistance. (Generically when we use a complex exponent we expect to see losses in the property we are measuring.)

Sometimes complex numbers are required and other times they are simply useful. And I know of no instance where complex numbers have been introduced for no reason...in fact in most calculations we avoid them where possible. For example there is no Quantum treatment of the infinite square well potential where we assume that the normalization coefficient is complex because we can achieve the same results by assuming it is real.

Other kinds of complex numbers show up all the time but they are buried in the Math terminology we use. For example the spin 1/2 operators actually fall into a representation of the "

quaternion group" where we have not one kind of imaginary unit, but three. i^2 = j^2 = k^2 = -1 where none of the i, j, k are equal.

On the one hand the imaginary unit is, well, imaginary. But it's so d*mned useful that I see no reason to not admit that such a thing really exists.

-Dan