Calculating Cross Power Spectral density between two complex signals

I have a small misunderstanding of cross power spectral density between two complex signals. I know that it is the Fourier transform of cross correlation between two signals. Let's say the complex signals are of the form (a+ib) and (c+id). Then the cross-correlation should result in something like

The real part result => ac - bd i.e. real1*rea12 - imag1*imag2;

complex part result => ad + bc i.e. real1*imag2 + real2 * imag1;

where real1, real2 represents the real part of both the signals and imag1, imag2 represents the imaginary part of both the signals.

But my professor said that the cross-correlation should result in

The real part result => ac + bd i.e. real1*rea12 + imag1*imag2;

complex part result => ad - bc i.e. real1*imag2 - real2 * imag1;

Where I am going wrong? Is my understanding of the equation form of cross-correlation correct? And also, I just want to know how to calculate cross PSD from this result. Can I just compute it like

Cross PSD result = |real_part result + imaginary_part result|.^2 ?

or Cross PSD result = |real_part result|.^2 + |imaginary_part result|.^2 ?

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.