If the Lagrangian is Hermitian, then fields and their complex conjugates are not independent. That is, you can solve the EOM for one field, and if you take the complex conjugate of that field, then that's how the complex conjugate field evolves: you don't have to solve the Euler-Lagrange equations for the complex conjugate field. If the Lagrangian is not Hermitian, does it even still make sense to say that fields are complex conjugate to each other, because then it'll no longer be true that one field is the complex conjugate of the other at a later time? I ask this because it seems that in Euclidean space, the Dirac Lagrangian is not Hermitian, yet the fields are denoted by the same symbol ψ, except the other field has a dagger. That doesn't seem to make sense, since if they are independent, they shouldn't be the complex conjugate of each other. Should there really be two independent symbols?