Detection complex vectors by using correlation
The reason we often use absolute value is that the received signal usually has unknown phase.

Consider the case in which $\boldsymbol{y=x_i}\exp(j\theta)$, where $\theta$ is a phase rotation.  The phase rotation is due not to "fading" but due to the fact that the received phase in communication systems is generally arbitrary due to the fact that the receiver's oscillator does not have the same phase as that of the received signal. (Note I ignored noise for the sake of simplicity, but this doesn't change the basic point.)

Now $\boldsymbol{y^Hx_i} = \boldsymbol{x_i^H} \exp(-j\theta) \boldsymbol{x_i} = \|\boldsymbol{x_i}\|^2 \exp(-j\theta)$.  Depending on the value of $\theta$, the real component of this term may be large or small.  That is what Stanley Pawlukiewicz was pointing out.  In his example, $\theta = \pi$, so the output of the correlation would be $-\|\boldsymbol{x}\|^2$.  The real component would be small, and by using the real component, one would erroneously conclude that the two signals were not aligned, when it fact they were aligned.  Using the absolute value corrects that problem.