In classical mechanics, it is often possible and convenient to describe a system with an object called a Lagrangian (in that it governs a system's behaviour, the Lagrangian is similar to a Hamiltonian). Like the Hamiltonian, the Lagrangian ought to be real - and any terms inside the Lagrangian ought to be Hermitian.
In quantum field theory (QFT), the kinetic energy, masses and interactions between fundamental particles (electrons, photons, quarks etc) are also described by a Lagrangian. There is a one-to-one correspondence between possible interactions and terms (or "operators") in the Lagrangian. When describing particles with a Lagrangian, we must write all allowed operators (i.e. interactions) in the Lagrangian. Some operators are forbidden by symmetries (e.g. charge consveration - by Noether's theorem, symmetries result in conservation laws).
The operator corresponding to a particle changing into an antiparticle is Hermitian, so on that basis is permitted in the Lagrangian, but in many cases, such an operator would violate a symmetry.
Clearly, an electron turning into an positron would break charge conservation, and thus the associated symmetry. Thus, we were forbidden from writing it in our list of interactions in the Lagrangian. This is an example of a selection rule - whilst it is possible to write a particle->anti-particle Hermitian operator, the operator is forbidden by symmetries. It is only possible if the particle has no quantum numbers, such that no symmetries are broken by particle->antiparticle, because particle=antiparticle.
The charge conjugation operator you refer to doesn't represent a physical process; it represents a way to (mathematically) replace particles with antiparticles in a mathematical theory. If a theory does not change under charge conjugation, the physics it describes would be the same if we replaced anti-particles with particles everywhere in the Universe. Remarkably, nature does not have this property.