You don't even need a compressor, water dissolves CO2 pretty well at STP. A bubbler will do, at least if you don't need much concentration.
But that really is the rub, isn't it? If you have low concentration, then a lot of water will evaporate in the process (indeed, more than the CO2 absorbed, I think, in most places on Earth!) while recovering not very much CO2; and its purity will be low (because O2 also dissolves in water, and to a small extent, Ar and N2).
And how much power are you going to spend, doing all of this? Water isn't free, you'll need a lot of it to capture much CO2. Or spend even more power condensing it. Every opportunity you have, to turbo-expand gas that's been compressed, or heat exchangers for fluids you're thermally cycling, must be taken. CO2 is a waste product as-is, so it better be minimal!
And indeed, as it happens, cleverer minds than mine have pondered this task. The choice of solvent is one with a fairly high boiling point (low vapor pressure: little lost to evaporation), and with a high affinity for CO2. Thus the concentration is fairly high, and the yield on extracting it is also high (say by thermal cycling). Ethanolamine seems to be a typical choice.
And, obviously it helps to start with a high CO2 source, like a coal plant; this can pretty much be compressed directly, and whatever doesn't liquefy as CO2 (which does need to be chilled enough to condense; CO2 is supercritical* very near room temperature) can just be vented (mostly N2 and Ar?). Liquid CO2 is a fantastic solvent, so some purification is probably a good idea (sheesh, does anyone distill liquid CO2? that's one high-pressure column..), but a typical proposal is to just pump the stuff to the bottom of the ocean, where the pressure keeps it in liquid phase, and being denser than water, just kinda... sits there, eventually diffusing into the water column, or the sea floor, or "freezing" as clathrates, for indeterminate consequences... well, someone's thought about it I'm sure, I just haven't read about it.
*Supercritical gasses mix the properties of liquids and gasses. They cannot be condensed by reducing temperature (at constant pressure), or by increasing pressure (at constant temperature). Basically the density of liquid and gas phases coincide at the supercritical point; the vapor is as dense as the liquid. So, uh, clearly this takes pretty high pressures, eh?
This looks like a pretty good overview for further reading:
https://www.cmu.edu/epp/iecm/documentation/2019Jan_IECM%20Amine-based%20CO2%20Capture.pdfTim