Refrigerator compressors and air conditioning compressors are both used to keep things cool. However, they work very differently. Refrigerator compressors use a refrigerant that is cooled by the coils in your refrigerator until it changes into a liquid and becomes a gas again when it enters your fridge. Air conditioners on the other hand use an electric compressor that compresses the refrigerant until it reaches its boiling point at which point water vapor will be released into the atmosphere through a condenser coil where the heat is removed from it by either air or water circulation before going back up to be compressed again. Which system you prefer really depends on.
Air purifiers are used to eliminate polluted air including dust smoke odour pollen hair etc. Bakery filters eliminate 99.9% bacteria 0.9-micron allergen including dust, pollen, etc. Whereas, the diffuser is used to eliminate oil in the air and spreads natural air fragrance to the environment.