Confusion About Complex and Scalar Fields

I have been reading QFT for the Gifted Amateur, on page 110, it reads:

"For the case of the real scalar field…it is Hermitian. For our
complex scalar field there is no reason why the field operator should be
Hermitian."

Is this because, the Lagrangian Density for a real scalar field doesn’t have any adjoints, and so when trying to use to calculate the Hamiltonian Operator (which is Hermitian), the scalar field would also need to be Hermitian? Whereas for a complex scalar field, it’s Lagrangian Density has adjoints in it and so it is not required for the field to be Hermitian, as the no-Hermitian components can be ‘cancelled out’, and so the Hamiltonian operator would be Hermitian?

EDIT: What I mean is, the Lagrangian density for a complex scalar field would be something like:
$$ L=(∂^{μ}ψ)^{†}(∂_{μ}ψ)-m^{2}ψ^{†}ψ $$
Since it is guaranteed that all the terms are Hermitian (since ψ† ψ is Hermitian), then there is no need for the field to be Hermitian. However, the Langrangian for a real scalar field, would be something like:
$$L=frac{1}{2}(∂_{μ}ϕ)^{2}-frac{1}{2}(mϕ)^{2}$$
Since it isn’t guranteed that the terms will be Hermitian due to the lack of terms adjoints, ϕ must be Hermitian.

My Question is: Is this reasoning correct?