Everyone knows that this is needed to make eigenvalues real, but still why we enforcing such a structure at first place? An arbitrary operator can have as complex as real eigenvalues, we can simply throw away the complex one by claiming they are not physical.

I did some search about this, and it seems, alike what books usually claims, the real reason hidden in the fact that the eigenvalues should not be just real, but bounded from below, otherwise there will be no stable quantum systems.

Well, the last paragraph I concluded from reading multiple claims here and there, if that is true, can anybody elaborate this in a more rigor/strict way? If wrong please answer on the first part.