When choosing compressor you should consider if it can constantly provide desired pressure and level of air. Little airbrush compressors small compressors boast impressive pressure figures but really poor flow rates enough limited to airbrush pistols. Also if small compressor can push enough air it will likely overheat in short while.
Our laser cutter came with small membrane pump (Smallest upon the picture), I really do not know what specifications it has but blast of air was extremely weak. It is sufficient if all you want to accomplish is engrave, although also then we got significant smoke traces around engraved picture.
it was rated for level of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It were able to give us continuous 1 bar (14 PSI) of pressure, Air Compressor For Laser Cutting nonetheless it had to operate constantly and overheat badly within 5 minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon container which is filled to ~100psi. With this unit we are able to run at constant ~2bar (31psi) without fear of overheating since compressor engine is not on continuously. Using reserve air flow in tank we can also push up to 3.5bar (50psi) for short jobs.
The best setup for me would be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via 
solenoid valve.