Schrödinger's Equation states that: $$i\hbar \frac{\partial \psi(x,t)}{\partial t}=H \psi(x,t).$$
Now the Hamiltonian operator, $H$ is hermitian as long as the potential $V(x)$ is a real valued function of position only. And when $V(x)$ becomes a complex valued function, or depends on $\hat p$, $H$ is no longer hermitian.
But, on the other hand, $i\hbar \frac{\partial}{\partial t}$ is anti-hermitian throughout, as $i$ is anti-hermitian and $t$ is a parameter only.
Why is it that $2$ operators on two sides of an equation, $H$ and $i\hbar \frac{\partial}{\partial t}$ performing similar operations on $\psi$ but exhibiting different hermitian behaviour? This happens to go against my intuition.
Is this thing normal by any means?
Correct me if I am wrong in my argument.
