I see a lot of books/lectures about classical field theory making use of complex scalar fields. However why complex fields are used in the first place is often not really motivated. Sometimes one can read as a footnote that a complex field is in principle mathematically equivalent to two real fields (see also What is the difference between complex field and two scalar fields?), but then the author often goes on using a complex field anyway.
This is confusing, because from quantum mechanics one learns that a complex quantity is not measurable. This is of course not the case in classical field theory, where both the real and the imaginary part must be simultaneously measurable quantities.
I heard physically motivated reasons for using complex fields like:
A complex scalar field represents different particles than a vector of two real fields. But this argument doesn't make sense in classical field theory, it is (if at all correct) only relevant in quantum field theory.
Only a complex field can represent charged particles, real fields are necessarily neutral.
A complex scalar field is a scalar and so it is by definition Lorentz invariant. A vector of two real fields is not Lorentz invariant and so one must use a complex field.
But I'm unsure which of these reasons (if any) is really valid. What is the point of using complex fields in classical field theory?