Consider
\begin{align} &\hat{e}_{\pm}=(1,\pm i,0) &\hat{z}=(0,0,1) \end{align}
Then $(\hat{e}_{\pm})^*\cdot (\hat{e}_{\mp})=0$ and $(\hat{e}_{\pm})^*\cdot \hat{z}=0$, so they are mutually orthogonal.
The cross product is
$$ \hat{e}_{\pm}\times \hat{z}=\pm i \hat{e}_{\pm}$$
So the result of the cross product is not orthogonal to one of the two vectors in the cross product. That means that the right hand rule is not valid for complex vectors?
