Is there a field which studies vectors of complex numbers or vectors of hypercomplex numbers in general? Is it useful to think of a mono audio signal as a vector of complex numbers where imaginary part encodes the timing information? Similarly, is it useful to think of a grayscale image as a quaternion vector where imaginary components encode the pixel position? Can this kind of logic be fruitful in any way?
Source: https://arxiv.org/abs/2006.08321
Not a dedicated field as such; the machinery for $\Bbb C^n$ or $\Bbb H^n$ is basically the same as for $\Bbb R^{jn}$, where $j\in\{1,\,2,\,4\}$ depending on what you're doing, but with some slight adjustments such as inner products being sesquilinear rather than bilinear.
Or even as a complex-valued function, depending on whether you'll use a Fourier series or Fourier transform to switch from time space to frequency space (or vice versa).
It's neither useful nor fruitful. At least with audio signals complex numbers help encode phase information and facilitate the solving of relevant differential equations. But quaternions are only worth using if you'll multiply them, because the only difference between $\Bbb H$ and $\Bbb R^4$ is that we define a multiplication operation on the former. Are you going to multiply two colours to get a third colour? I don't think so. In fact, the quaternion you get won't even be part of your $3$-dimensional subset in general.
You could do something like this with CMYK, provided you use a coordinate transformation that lets each component be unbounded, but it'd still be pointless. If for example you learn yellow times pink is green (which wouldn't even stay true if you switched to another transformation), is that useful? No, not at all.