Why are complex numbers needed to understand Quantum Mechanics?

Question

I kind of understand that complex numbers were invented to solve the equations of type x^2 + 1 = 0. However, I do not understand the need of complex numbers in understanding / modeling quantum systems. Quantum systems are natural and nature does not function in complex numbers OR at least I can't see it that way. Then why do the quantum systems need complex numbers?How can I "reverse engineer" a system of quantum state expressed in complex numbers to its actual natural manifestation (assuming some kind of mechanical tools are developed in the future to do that)?

nabla9 · Accepted Answer

> nature does not function in complex numbers
It's the opposite in fact. Almost everything in the nature works more naturally with with complex numbers. Complex numbers are algebraically closed unlike real numbers as you point out and that is very important part of it. Everything becomes more difficult if you try to force them into real numbers.
Quantum wave function is an explicitly complex object. Making it anything else makes it more difficult.
Here is great and easy to understand explanation: https://www.scottaaronson.com/democritus/lec9.html

fsflover · Answer

This is simply how quantum mechanics works. Schrodinger equation involves complex numbers. To remove them from the final result, we take modulus square of the amplitude and obtain the probabilities. Why it works is an open question in physics AFAIK.

transfire · Answer

Its not necessary to use complex numbers, its just convenient. Look into Euler's Identity.