Yes, the same approach can be used for complex matrices, with the constraint becoming
$\left[ \begin{array}{rr} W_{1} & X \\ X^{H} & W_{2} \\ \end{array} \right] \succeq 0 $
Here, the constraint says that the matrix must be Hermitian (rather than symmetric) and positive definite.
However, not all SDP solvers directly support complex matrices. You can reformulate any SDP involving complex Hermitian matrices into an SDP with real symmetric matrices. A constraint of the form
$Z \succeq 0$
where $Z$ is complex and Hermitian becomes
$\left[\begin{array}{rr} \mbox{real}(Z) & -\mbox{imag}(Z) \\ \mbox{imag}(Z)^{T} & \mbox{real(Z)} \\ \end{array} \right] \succeq 0 $
The resulting problem has a matrix variable of size $2n$ by $2n$ but this doesn't actually require any additional storage, since each complex number in $Z$ requires twice as much storage as each real in the reformulation.