Complex numbers are just a way of representing an additional degree of freedom in an equation. You have to represent complex numbers not on a number line but on the complex plane, so each complex number is associated with two numbers. That means if you create a function that requires two inputs and two outputs, you could "compress" that function into a single input and output by using complex numbers.

Complex numbers are used all throughout classical mechanics. Waves are two-dimensional objects because they both have an amplitude and a wavelength. Classical wave dynamics thus very often use complex numbers because you can capture the properties of waves more concisely. An example of this is the Fourier transform. If you look up the function, it looks very scary, it has an integral and Euler's number raised to the negative power of the imaginary number multiplied by pi. However, if you've worked with complex numbers a lot, you'd immediately recognize that raising Euler's number to pi times the imaginary number is just how you represent rotations on the complex plane.

Despite how scary the Fourier transform looks, literally all it is actually doing is wrapping a wave around a circle. 3Blue1Brown has a good video on his channel of how to visualize the Fourier transform. The Fourier transform, again, isn't inherently anything quantum mechanical, we use it all the time in classical mechanics, for example, if you ever used an old dial-up model and wondered why it made those weird noises, it was encoding data as sound wave by representing them as different harmonic waves that it would then add together, producing that sound. The Fourier transform could then be used by the modem at the other end to break the sound back apart into those harmonic waves and then decode it back into data.

In quantum mechanics, properties of systems always have an additional kind of "orientation" to them. When particles interact, if their orientations are aligned, the outcome of the interaction is deterministic. If they are misaligned, then it introduces randomness. For example, an electron's spin state can either be up or down. However, its spin state also has a particular orientation to it, so you can only measure it "correctly" by having the orientation of the measuring device aligned with the electron. If they are misaligned, you introduce randomness. These orientations often are associated with physical rotations, for example, with the electron spins state, you measure it with something known as a Stern-Gerlach apparatus, and to measure the electron on a different orientation you have to physically rotate the whole apparatus.

Because the probability of measuring certain things directly relates to the relative orientation between your measuring device and the particle, it would be nice if we had a way to represent both the relative orientation and the probability at the same time. And, of course, you guessed it, we do. It turns out you can achieve this simply by representing your probability amplitudes (the % chance of something occurring) as complex numbers. This means in quantum mechanics, for example, an event can have a -70.7i% chance of occurring.

While that sounds weird at first, you quickly realize that the only reason we represent it this way is because it directly connects the relative orientation between the systems interacting and the probabilities of certain outcomes. You see, you can convert quantum probabilities to classical just by computing the distance from 0% on the complex plane and squaring it, which in the case of -70.7i% would give you 50%, which tells you this just means it is basically a fair coin flip. However, you can also compute from this number the relative orientation of the two measuring devices, which in this case you would find it to be rotated 90 degrees. Hence, because both values can be computed from the same number, if you rotate the measuring device it must necessarily alter the probabilities of different outcomes.

You technically don't need to ever use complex numbers. You could, for example, take the Schrodinger equation and just break it up into two separate equations for the real and imaginary part, and have them both act on real numbers. Indeed, if you actually build a quantum computer simulator in a classical computer, most programming languages don't include complex numbers, so all your algorithms have to break the complex numbers into two real numbers. It's just when you are writing down these equations, they can get very messy this way. Complex numbers are just far more concise to represent additional degrees of freedom without needing additional equations/functions.