I wonder whether the complex conjugation operator, defined on a wavefunction as
$$ C \psi(x) = \psi^*(x), $$
On one hand, its eigenvalues are not necessarily real. On the other hand, since $C$ is an antilinear operator, its adjoint must satisfy
$$ ⟨u|Cv⟩ = ⟨C^\dagger u|v⟩^*. $$ In $x$-space this would be
$$ \int dx u^*(x)Cv(x) = \bigg[\int dx [C^\dagger u(x)]^* v(x)\bigg]^*. $$ But since the LHS is just $\int dx u^*(x)v^*(x) = [\int dx u(x)v(x)]^*$, $C^\dagger$ must be $C$ itself.
So the question is whether $C$ is Hermitian, or, more generally, how do we define Hermiticity for antilinear operators?