⟨a_\[VerticalSeparator]b_⟩:=Conjugate[a].b

Conjugate[1,2,3]

Unprotect[Conjugate];
Conjugate[a_,b_,c_] := a+b+c;
Protect[Conjugate]

Conjugate[1,2,3]

Conjugate[2,3,4]