Why complex signal has no imaginary spectrum
The imaginary part of the spectrum corresponds to the odd part of the time-domain sequence. Since the given signal is even, i.e.,

$$x[n]=e^{jn\omega_0}=x^*[-n]=e^{-j(-n)\omega_0}=e^{jn\omega_0}$$

the imaginary part of the spectrum is zero, i.e., the spectrum is purely real-valued.

Note that for complex-valued signals, even in this context means $x[n]=x^*[-n]$, i.e., its real-part is even and its imaginary part is odd.

In sum, if a sequence $x[n]$ satisfies

$$x[n]=x^*[-n]$$

(i.e., the sequence is even and, consequently, its odd part is zero), then its DTFT is purely real-valued.