  A bounded linear operator $T$ on a separable complex Hilbert space $H$ is
called $C$-normal if there is a conjugation $C$ on $H$ such that $ CT^\ast
TC=TT^\ast$. Let $\varphi$ be a linear fractional self-map of $\mathbb{D}$. In
this paper, we characterize the necessary and sufficient condition for the
composition operator $C_\varphi$ and weighted composition operator
$W_{\psi,\varphi}$ to be $C$-normal with some conjugations $C$ and a function
$\psi$.
