Amplify a signal and phase shift it by multiplying by a complex number
Multiplying by a complex number causes a rescaling by the magnitude and a rotation by the angle.  This is why when you multiply two complex numbers, you multiply the magnitudes and add the angles.

The only signal for which a complex multiplication (rescale and rotation) is the equivalent of a phase shift is a single complex pure tone (looks like a spring, or threads on a bolt).  The frequency is the translation value between the rotation amount and the shift size in the time domain.  Thus if you have a signal that is a sum of complex pure tones with different frequencies, each will be shifted by a different amount by the same non-real multiplication factor, and the sum of the shifted tones will not match a shift of the sum.

Since a real pure tone is the sum of two complex pure tones of different frequencies, it cannot be shifted by a complex scalar multiplication, nor can any sum of them.

That's all there is to it.