functions - Wrong behaviour of ComplexExpand and Conjugate

This input gives the wrong result

ComplexExpand[Conjugate[f[x]], {x}]
(*f[x]*)

instead of Conjugate[f[x]].

In particular it works well with some functions, e.g.

ComplexExpand[Conjugate[f[x] /. f -> Sin], {x}]
(*Cosh[Im[x]] Sin[Re[x]] - I Cos[Re[x]] Sinh[Im[x]]*)
ComplexExpand[Conjugate[f[x, y] /. f -> Times], {x, y}]
(*-Im[x] Im[y] + Re[x] Re[y] - I (Im[y] Re[x] + Im[x] Re[y])*)

(note that ComplexExpand[Conjugate[f[x]], {x}]/. f -> Sin doesn't work) but not with others e.g.

ComplexExpand[Conjugate[Dot[x, y]], {x, y}]

(*x.y*)

Any idea how to deal with this problem? How can I get the correct output for the last example f=Dot? and for a generic function f?

Why the functions Dot and Times behaves in a different way?