EA-50 Captain reported receiving an anti-skid system failure alert while on final; and may have had a potential NMAC event with another aircraft.

Narrative

The C172 in question and I had each other in sight and were communicating. We were on right base with apparent equal distance to final. Upon extending my landing gear I had an anti-skid system failure alert. Not an emergency - however; asked the other plane for priority to land with this problem. The other pilot said 'no' to my priority request and stated; 'You just think you should have priority because you are faster;' or words to that effect. I then stated that I need to land immediately. I asked him to go around several times… another aircraft obliged and made a 360; but not the aircraft in question. We made our turn to final while still communicating back and forth… I had him in sight and continued. The other aircraft flew over top of us but we both had each other in sight and were in constant communication. Due to being a dark night; I saw the lights of the other aircraft but was not able to judge the distance. I continually stressed that I needed to land immediately and was going to get to the airport first. Both aircraft maintained visual and radio communications throughout. The other pilot was belligerent and uncooperative throughout. I was flying an Eclipse jet with passenger witnesses.