Coming from someone who knows a tiny bit about the subject but who really wants to learn. I know it's the square root of -1 but I would like some insight as to why it's used at all.
As @AccidentalFourierTransform pointed out, the use of complex numbers isn't strictly necessary to describe quantum mechanics. It's simply the first (and in a sense, least complicated) mathematical structure people learn that is able to describe quantum mechanics properly.
For an easy example of quantum mechanics entirely using real numbers, we simply note that the map $f$ from complex numbers to invertible 2x2 matrices given by
is an isomorphism. This means we can effectively replace all of the complex numbers in quantum mechanics with these matrices, which contain entirely real entries.
Quantum mechanical models use solutions of wave equations. These are second degree differential equations which accept solutions using the complex number formalism.
This is not particular to quantum mechanics. Electromagnetic radiation is described by solutions of the maxwell equations and are also formulated with the complex numbers formalism.
It is the quantum mechanics postulates that define the difference between classical wave solutions and quantum mechanical, not the i.
Whereas for classical equations the variables in the solutions are measurable in the laboratory , as for example electric and magnetic fields, in quantum mechanics it is the complex conjugate squared of the complex wavefunctions that have a connection with measurable quantities, describing probability distributions for the observables.