Does it make sense to have complex numbers representing real-world audio signals?
So it seems like real-world (discrete) audio signal might have complex values when being represented digitally, 


No, you misunderstood that. The discrete audio time signal doesn't have non-real values. The Fourier transform can have such.


  but does this make sense?


It doesn't need to make sense. It's just math. I sometimes remind myself of that – it helps to occasionally get some distance from the very human desire to put sense into things.

However, assume any signal has Fourier transform $S(f)$, and that transform is purely real. Now you shift that signal in time by $\Delta t$. From knowing the Fourier transform properties, or by just applying the integral (continuous FT) or the sum (discrete FT), you instantly see that this means multiplication with a $e^{j2\pi \Delta t f}$. In the general case, this produces non-zero imaginary parts where there were none before.

In other words: even if you have a signal whose FT is purely real, simply by shifting it in time a bit, you can always get a non-zero imaginary part in the spectrum. That applies to all signals, be it audio, or I/Q baseband of a radio receiver.


  If yes, then how can we interpret this?


I think it makes sense for you to just continue to follow these lectures, I don't want to spoil too much, but:

only time-symmetrical (to be exact, time-hermitian, but since audio signals are real, that's identical to symmetry) signals have a real spectrum. If you see a non-real spectrum, you can simply tell the signal wasn't hermitian (symmetric).

Also, it tells you that if you want to deal with the spectra of signals, you always have to consider both, amplitude and phase, to represent a signal. And neither of these two aspects can be defined on the real part of the spectrum alone! The first is the root of the sum of squares of the real and the imaginary part (so the imaginary part is important), and the second is the arcustangens of the ratio of imaginary and real part. Phase "contains" the time-shift, measured in full signal periods, for harmonic signals, by the way, which is a class of signals very important for Audio. 


  Isn’t the value of the input supposed to represent change in air pressure?


yes. But that the time-domain signal, not the frequency-domain signal.


  So if for an input signal x, say x[0] was 2 + 3j, what would this tell us about the pressure on the diaphragm at that point of time?


You're confusing a real time-domain signal with its Fourier transform. A single scalar value, like the momentary air pressure, is real. there's no $+3j$; there can't be, in time domain. However, in frequency domain, there might be an imaginary part.

Make sure that you understand this: 

A signal is just a changing value over time, or some other axis.
The most intuitive representation hence is one along that axis, for example, an audio signal might be a change in pressure over time.

However, there's other ways of expressing the same signal. The Fourier Transform of a signal describes the same signal, but is a different function than the time signal. We often speak of "frequency domain" (after FT) and "time domain".

I don't know your background, but early on in my studies, we were taught of vector spaces, bases and base transforms.

"Time steps" is just one basis with which you can describe a signal; "Fourier Coefficients" is another one. The same signal has different coefficients (ie. representations as vectors of numbers) regarding these two bases. And the discrete FT is really, really, really nothing but a plain, old, boring base transform matrix with a few nice properties!