For any real signal in time, the spectrum (consisting of coefficients of $e^{j\omega t}$) will be complex conjugate symmetric (positive and negative frequency components with the same magnitude and opposite phase for any given frequency). When the spectrum is not complex conjugate symmetric, then the time domain waveform MUST be complex: every value in time will have a magnitude and phase or similarly a real and imaginary component.
This is a valuable property for use in digital wireless communications, where we are often motivated to get as much data rate between a transmitter and receiver in the minimum amount of bandwidth. If we up-convert a real baseband signal to a carrier frequency, those symmetric positive and negative frequencies as baseband will map to the upper and lower sidebands of the carrier. For the case of a complex baseband signal where the positive and negative frequencies can be independent, we can get twice the data rate in the same amount of bandwidth with all else equal over the case of a real baseband signal.