C172 pilot reported being notified of another aircraft also on base that almost went unnoticed during a final approach at a non-towered airport. The two aircraft increased separation and the reporter proceeded to land.

Narrative

I entered the traffic pattern for Runway 34 from the east. I was at 1350 ft. MSL; monitoring / communicating CTAF. After scanning for traffic; I called entering downwind at midfield and then executed what I called. One aircraft reported left crosswind 34; so they were assumed to be no factor. In hindsight; they may have been turning left base but made an incorrect radio call. I turned base after scanning and announcing my intentions on CTAF. As I was configuring for landing; a voice came on the radio asking if the two aircraft on base had visual on each other. I made some shallow clearing turns to check behind my aircraft's blind spots. I found the second aircraft 300 ft. below and to my right; obscured behind my nose originally. At that point; we discussed a plan; the other aircraft then climbed up and to the right while I went left and continued with my landing. I guess technically I cut them off by standard right of way rules; but their presence in the traffic pattern was unknown. The conflict was resolved without issue; and the second plane came and landed behind me.Potential contributing factors. Other aircraft made improper or no radio calls. Other aircraft was brown and blended into the terrain; from my perspective. I crossed midfield at traffic pattern altitude. PAQ has turboprop traffic; so I tend to avoid the AIM (Aeronautical Information Manual) 'cross airfield at 500 ft. above traffic pattern altitude' suggestion since that would put me at traffic pattern altitude for the big fast guys. Other aircraft was likely below recommended traffic pattern altitude. I estimate they were 300 - 500 ft. below me when I saw them. PAQ is a busy airport with lots of flight training taking place. We regularly get 3 taxiing for departure; 7 in the pattern; and 5 inbound to land. It is congested and busy which can lead to some overload; laziness; or complacency in the traffic pattern. Sometimes people do nonstandard things to try and get down a little faster. PAQ is in the middle of construction; so taxiways are open and closed sporadically. This also leads to delays in the air as people back-taxi on the runway.