I use this question to point out the confusion we get when conflating phase with time delay once working with signal processing using complex waveforms, and to make it clear with complex signal processing such as the Fourier Transform, any of the bins can have a magnitude and phase, including the "DC bin". Often baseband equivalent signals are used where the carrier becomes "DC" facilitating further analysis and simulation. I also use this to make it very clear that we need two data-paths to implement complex baseband signals in practice (just as we need two real numbers to represent a complex number on paper!). 

A big take-away when working with complex signals is that phase is a rotation on the complex plane (I recommend not associating it with two sine-waves offset in time-- it's true that such a delay will cause the rotation I describe, but those stuck in that thinking will believe then that it is impossible to create in hardware a signal that is constant with time that also has phase). A time delay results in a linear phase in frequency, but that is not the only way to induce a phase rotation. 

A "DC signal with phase" is used in practice anywhere we want to add a static phase offset (the "static" part makes it DC, as in not changing with time). One example of this is an RF phase shifter using a vector modulator such as the [Analog Devices HMC630][1] shown in the block diagram 
below:

[![HMC630][2]][2]

 A vector modulator has two ports labeled "I" and "Q", where "I" represents "In - phase" and "Q" represents "Quadrature". Each of the ports can take a DC signal, and therefore together represent a complex DC signal that we would apply as $I+jQ$.  The device will upconvert this complex DC signal to the RF carrier, which is the signal applied differentially to IN and /IN ports. Thus if we rotate our DC signal x degrees, x degrees will also be added the RF carrier's phase appearing at the RFOUT port. 



The above implementation when shown in complex form is functionally equivalent to this block diagram below, which (once understood) demonstrates the significant convenience of the complex representations:

[![Vector Modulator as a Phase Shifter][3]][3]

The "Hilbert" block creates the "[Analytic Signal][4]" and is implemented with a 90 degree quadrature splitter, which produces two outputs $\cos(\omega_c t)$ and $\sin(\omega_c t)$ which I represent above with the thicker line as a single complex datapath as $\cos(\omega_c t)+j\sin(\omega_c t)= e^{j\omega_c t}$. The multiplier shown above is a full complex multiplier, which would require four multipliers and two adders generally as a product of imaginary terms as  $(I_1+Q_1)(I_2-jQ_2)$, where $I_1$, $Q_1$ are the real and imaginary ports from the output of the Hilbert, and $I_2$, $Q_2$ is our "DC phase". Since we in this case only need the real of the result (as a passband signal), we only need two of the multipliers representing the real terms of the complete product: $I_1I_2 + Q_1Q_2$, matching the implementation shown.

 


  [1]: https://www.analog.com/en/products/hmc630.html
  [2]: https://i.sstatic.net/yfDMR.png
  [3]: https://i.sstatic.net/h96Ye.png
  [4]: https://dsp.stackexchange.com/a/80983/21048