Oil filter

I’ve never done that with my compressor, however I remember on a generator I had we used to drain the oil while it was still hot and run it through a paper coffee filter a couple of times. The old generator worked fine and since we always were adding fresh oil to make up for what was lost in the filtering no total changes were done except if smelled burnt.
 
It is impractical for many reasons. Here are just a few of them.

1. I doubt there is any manufacturer selling a filter that is cost effective for extending the life of 12 oz. of oil.

2. Automobiles and large continuous duty compressors circulate oil in a pressurized system. Small compressors use splash lubrication so the oil isn't being circulated through a filter even if there was one. Continuos duty compressors don't use filters either.

3. Automotive oils have detergent additives to suspend dirt and particles for the filter to collect. Compressors use non-detergent oil because they have no filter and detergents in oil are harmful to the compressor.

4. The filter would cost more than the 12 oz of oil it was there to protect.

5. After an initial 5 hour oil change, compressor manufacturers require annual oil changes or at 25-50 hours of use depending on the manufacturer. Percentage wise, very few air gunners run a compressor more than 50 hours a year so filters would just add to the maintenance expense as it would necessitate an oil and filter change. The reason for the annual oil change requirement is that oil will absorb moisture over a year regardless of the hours it runs. Water in oil will rust internal components if the oil isn't changed.