Calculating Cross Power Spectral density between two complex signals
Cross correlation is defined as $R_{xy}(\tau) = E\{x(t)y^*(t+\tau)\}$. You missed to take the conjugate of $x_2$. So $x_1x_2^* = (c+id)(a-ib) = ac+bd +i(ad-bc)$.

Cross PSD is the fourier transform of $R_{xy}(\tau)$, not the two equations you mentioned.