The question you should ask is what does it mean to have a complex signal. Once you get that you will know there's no need to ask for an example.

When you use an Oscilloscope you only see real signal cos(wt) or sin(wt), or in general A cos(wt + phi), the imaginary part does not exist. To make a stronger point imagine a mechanical system like a speaker cone. The position of the speaker is real, we can measure it. We have all the information to understand and analyse the system with real signals.

When you deal with complex signals, A exp(j(wt + phi)), is it a mathematical abstraction. It works because linear systems by definition must follow the rule of superposition.

The key points are:

- The complex signal contains the same information as the real signal. It's just a different representation of the same thing.

- The whole purpose of the mathematical abstraction is to make computations easier.

You can see why it holds together from these notes,

https://www.cs.ccu.edu.tw/~wtchu/courses/2012s_DSP/Lectures/Lecture%203%20Complex%20Exponential%20Signals.pdf If you have ever solved equations for electrical or mechanical systems you will know that it can be tedious. You are continually messing with both the cos part and the sin part and using trig identities like cos(A+B). The exponential form makes life a lot easier, that's why we use Laplace transforms and complex numbers to solve AC circuits, not differential equations.

There's even a real number equivalent of the Fourier transform. It is very slow to compute even for a computer. The invention of the FFT was a big step forward but it needs complex signals to see the patterns and make it fast.

The FFT produces complex results. That's where you get stuck. There's one step missing, the one which converts the complex signal back to the underlying real signal it represents.

You can summarize all the above processes as,

- Real signal

- Convert to complex signal equivalent

- Do processing or calculations

- Convert complex signal back to real result.

The complex numbers are just a stepping stone to make the calculations easier.