The quantum field resultant from the quantization of a real classical field is hermitian, but why the quantum field corresponding to a complex classical field should be non-hermitian?
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Note that the spectrum ${\rm Spec}(\hat{A}) \subseteq \mathbb{R}$ of a Hermitian/self-adjoint operator $\hat{A}$ belongs to the real axis $\mathbb{R}\subseteq \mathbb{C}$, cf. e.g. this Phys.SE post.
It is therefore not surprising that a reality condition for a classical field naturally translates into a Hermiticity/self-adjointness condition for the corresponding operator, and vice-versa.
