  We prove the following analogue of the classical Skitovich--Darmois theorem
for complex random variables. Let $\alpha=a+ib$ be a nonzero complex number.
Then the following statements hold. $1$. Let either $b\ne 0$, or $b=0$ and
$a>0$. Let $\xi_1$ and $\xi_2$ be independent complex random variables. Assume
that the linear forms $L_1=\xi_1+\xi_2$ and $L_2=\xi_1+\alpha\xi_2$ are
independent. Then $\xi_j$ are degenerate random variables. $2$. Let $b=0$ and
$a<0$. Then there exist complex Gaussian random variables in the wide sense
$\xi_1$ and $\xi_2$ such that they are not complex Gaussian random variables in
the narrow sense, whereas the linear forms $L_1=\xi_1+\xi_2$ and
$L_2=\xi_1+\alpha\xi_2$ are independent. We also study an analogue of the Heyde
theorem for complex random variables.
