"Where" exactly are complex numbers used "in the real world"?

Complex numbers are used in electrical engineering all the time, because Fourier transforms are used in understanding oscillations that occur both in alternating current and in signals modulated by electromagnetic waves.

I was asked this exact question by my wife last night. She was looking for an everyday example of the use of complex numbers to explain to her 8th grade math class (whose knowledge of complex numbers consists of $i = \sqrt{-1}$ ).

My response was this:

Imagine an electronic piano. Each key produces a different tone. A volume control changes the amplitude (volume) of all the keys by the same amount. That's how real numbers affect signals.

Now, imagine a filter. It makes some keys sound louder and some keys sound softer, depending on their frequencies. That's complex numbers -- they allow an "extra dimension" of calculation.

(Yes, I know about phase shifts and Fourier transforms, but these are 8th graders, and for comprehensive testing, they're required to know a real world application of complex numbers, but not the details of how or why. I don't understand this, but that's the way it is)

Since you mentioned "real world".

The "real world" consists of miniscule particles: protons, electrons, etc. Which are not exactly particles: quantum mechanics says each of them looks like a wave. Normal waves have some "value" or "displacement" or "magnitude" in each point of space.

Magnitude (amplitude) of waves in quantum mechanics are complex! Just imagine, the whole "real world", everything you can see or touch consists of some waves with complex amplitudes!

Complex numbers are used in real world literally EVERYWHERE.