C-182 pilot reported a NMAC with another aircraft on final approach at a non-towered airport.

Narrative

I was pilot and sole occupant of a Cessna 182C at ZZZ. The runway in use was XX with a left traffic pattern and there were multiple training aircraft in the traffic pattern including several Diamond DA-42 Twin Star's and several Cessna 172 Skyhawk's from multiple large; collegiate flight schools located at nearby ZZZ1. As I held short of Runway XX a DA42 announced a non-standard right base for XX. Several other aircraft in the pattern commented on the nonstandard pattern. After the DA42 had made a touch and go landing I entered Runway XX for closed traffic. As the DA42 ahead made a left crosswind turn; I began the takeoff roll. At 700 ft. AGL I announced and the made a left crosswind turn. Approximately 1/2 mile south of Runway XX and at 1000 ft. AGL I announced and made my turn to the downwind leg. I had several aircraft in sight ahead on the downwind. One aircraft was very wide (I would estimate a more than a mile from the runway and several hundred feet above the standard traffic pattern.) Another aircraft reported several miles east inbound on an instrument approach. As I approached what would have been my base turn; I saw another aircraft on a base leg about a half mile ahead and another on final approach an estimated 2 miles east. I announced that I had two prior aircraft in sight and would extend my downwind leg and be number 3 for landing. As I came abeam of the second landing aircraft; I announced and made my left base turn and began descending. While on the left base leg I heard another aircraft announce left base. I looked all around and was not able to see another aircraft. I announced and made my turn to final for Runway XX. At that point another aircraft reprimanded me over the CTAF for 'cutting me off'. I did not see a conflicting aircraft until a DA42 overtook me from above and announced/executed a go-around. There was some unprofessional 'back and forth' on the CTAF where; in the heat of the moment; we expressed our differing opinions of who was at fault. I landed without further incident. I still do not know where the other aircraft was prior to my turn to final. I suspect it was at my 4 or 5 o'clock position on the base leg. I thought I had made observations on the base leg; but as I had extended the downwind leg for traffic. I had assumed other traffic would be to my left. Clearly I was wrong. As a regular user of ZZZ I notice a high volume of training traffic from several local large flight schools. I understand that it is convenient field; with cheap fuel and is just over 50 NM from ZZZ1 making it a good cross country target. It is; however; also a pilot controlled field where a radio is not required and is charted and NOTAMed for intense parachute operations. I think safety would be enhanced if big flight schools limited their operations at such fields accordingly.