What is the physical meaning of a complex signal in the time domain?

While studying the Discrete Fourier Transform, the subject of a discrete time complex signal is raised, does it have any physical meaning?

It is used in radar and communication theory - I'll give you a radar example:
(I understand that it is sines and cosines that occur in nature)

Let's say a radar is going to be transmitting and receiving pulses of RF.
Forming pulses and modulating phase over a wide frequency range is problematical.
Oftentimes, the better solution is to use a fixed intermediate-frequency (IF).
So the pulse is formed at the IF, and mixed to the desired transmit/receive frequency.
Receiving demodulates by the difference frequency, and mixes back to the IF.
(a heterodyne - keeping in mind that at each mix the unwanted sideband is filtered)

From the IF, one can mix with the IF and a 90 degree phase-shifted version of itself,
and demodulate down to around DC. This is called in-phase, quaderture (IQ) demodulation.
Amplification and A/D conversion is then performed on the two orthogonal signals (cos & i sin),
and offers the potential for the greatest possible SNR processing gain to be achieved.

If I understand your question, you want to understand the physical meaning of complex-valued quantities, not just in DFT, but in general.

A mathematical model is a representation of a physical phenomenon or process, and idealized version of reality with emphasis on some characteristics of it. It uses the language of mathematics to describe the phenomenon under study. Mathematics offer several tools which allow us to process the model and get insight. A model must be detailed enough in order to reflect reallity accurately but not too complex in order to be able to use it.

Why we use mathematical models of real-life phenomena? Because: a) Mathematics is a universal and compact language. b) You can derive results and conclusions just by following the rules of mathematics in a way that no other representation (e.g., verbal description) allows. Sometimes you don't have even to understand intuitively the underlying processes; you only have to do the calculations and the mathematical manipulation and the result pops out!

While natural numbers are a part of physical world (animals have sense of counting, so do have babies, the sense of counting is inherent to all beings equiped with some kind of logic), mathematics as a scientific field is a construction of our minds. However, it has numerous applications in real-life, under some conditions: a) We use the proper model for the current situation of interest, and b) we interpret the results in a proper manner.

And now we come to the complex numbers. It turns out in many cases that if we accept the existence of complex numbers or, to simplify it, of imaginay numbers, that is, if we assume that the square root of negative numbers exist, then: a) many unsolved problems are solved, b) the required computations in some other problems become suprisingly easy, and c) in some other problems we get insight which no other mathematical tool offers! Finally, in some cases (like in Fourier series) it is just a way to use a more compact notation.

One may ask: are complex numbers "real"? Well, besides the philosophical extensions in our discussion of "what do you mean by real", let's just say that: it is real in terms that it has real-life applications. But, at the end, it is just an intermediate step towards the final solution of our problem, which will always be some real-valued quantity, because complex-valued quantities have no physical existence. Example: In digital modulation, we often have complex-valued symbols that modulate a sinusoidal carrier. In practice, the real part modulates a cosine wave (in-phase component) and the imaginary part modulates a sine wave (quadrature component) - this is the so-called I/Q modulation. The use of complex-valued symbols is an easy way to get insight and to describe this communication technique. If we interpret the result correctly (that is, we take the real part and the imaginary part which are real-valued quantities and use these to modulate two quadrature carriers), then we will have a real-life application.

The way I see it, it is like a different interpretation of a phenomenon. For example, you can describe gravity either as a force field, as a "distortion" of space-time, as a oscilation of higher dimensions in our 4-D (including time) reality, and so on. What is real? Nothing! As long as they work in practice, they are all real, and at the same time, no one description is real! It is mathematics, and even though mathematics have applications in real life, they are just a construction of our mind ...

All these are just the way I see the subject, I am sure many people will disagree. I would love to have this conversation, especially with people that are experts in the field, because I am not a mathematician or a physicist. It is just the way I see it.