When choosing compressor you should consider if it will be able to constantly provide desired pressure and level of air. Small airbrush compressors little compressors boast impressive pressure figures but actually poor flow rates adequate limited to airbrush pistols. Also if small compressor can push enough air it will likely overheat in couple of minutes.
Our laser cutter was included with little membrane pump (Smallest upon the picture), I really do not know what specifications it has but stream of air was extremely weak. It is enough if all you have to to accomplish is engrave, although even then we got substantial smoke traces around engraved picture.
it was rated for level of ~2.4cfm and had a 1 gallon Air Compressor For Laser Cutting container which may be filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, but it had to run constantly and overheat badly within five minutes.
Third and current compressor we got was 6cfm rated belt driven unit with 13 gallon tank which is filled to ~100psi. With this unit we are able to run at continuous ~2bar (31psi) without concern with overheating since compressor engine is not on continuously. Using reserve air flow in tank we can also force up to 3.5bar (50psi) for brief jobs.
The best setup in my opinion will be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.