  We show that for any fixed $\theta\in(-\frac{\pi}{2},\,0)\cup
(0,\,\frac{\pi}{2})$, the 1-dimensional complex Ornstein-Uhlenbeck operator
  \begin{equation*} \tilde{\mathcal{L}}_{\theta}= 4\cos\theta
\frac{\partial^2}{\partial z\partial \bar{z}}-e^{\mi\theta} z
\frac{\partial}{\partial z}-e^{-\mi\theta}\bar{z} \frac{\partial}{\partial
\bar{z}}, \end{equation*} is a normal (but nonsymmetric) diffusion operator.
