[wwarner]

You might think that in every real world application, complex numbers are introduced as a convenience, and that every calculation that takes advantage of them ends with taking the real part of the result, but that's not the case. In QM, the final answer contains an imaginary part that cannot be removed.

[BeetleB]

Having studied QM, I disagree. You can always introduce trigonometry and remove it.

Keep in mind that everything observable is a real number even if the intermediate calculations involve imaginary numbers.

[tome]

Yup, and the Madelung equations explicitly present QM without complex numbers:

https://en.wikipedia.org/wiki/Madelung_equations#Equations