What is the purpose of complex conjugation in quantum physics? What is it that complex numbers allow us to do that can't be done otherwise, or at least cannot be done as easily? I understand what complex numbers are and how complex conjugation is done, yet I can't find a straightforward explanation of the role it plays and why quantum equations were framed in that context in the first place. As simple example, at http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/psi.html we find "In order to keep this topic as simple as possible, we're going to start by living in a very simple universe. Our universe has only three points: x=1, x=2, and x=3. Our particle must be on exactly one of those points. It cannot be anywhere else, including in between them. Since those three points define the whole universe, the wavefunction itself is defined at all three of those points, and nowhere else. So Y is just three complex numbers, which might look something like this. " . . . and then lists the numbers. But why complex numbers?