Nature follows differential equations.

Differential equations are generally complex to solve but have properties that are helpful. E.g., a second order differential equations would have two independent constants of integration in the solution. If two have already been found, a third independent one won't be needed and you already know that you have a complete solution.

For linear equations, if you find enough orthogonal basis functions that are each a solution, even if by hook or crook, you would have found all solutions.

Some functions and operators yield a function with the same form as the original function. E.g., derivative of exponentiation, second derivative of sine, etc.

The differential equation can happen to be such that a function of the above type then cancels out from the equation. This simplifies the equation.

If the above process yields enough orthonormal basis functions, then we know we have the entire space of solutions.

When the operator happens have some properties, the above happens.

Euler's equation links exponential to sine and cosine. That makes complex numbers useful. Then exponentiation covers sine and cosine too.

Magnitude of complex numbers is the number multiplied by its complex conjugate.

The math proofs extend from real numbers (actually needed by Physics) to complex numbers by using complex conjugates.

Hermitian matrices and self-adjoint operators are special cases that bring orthogonal basis functions, real-valued solutions (in spite of using complex numbers), etc.

References:

Spectral Theorem

Sturm Liouville Theory