Instructor pilot reported executing an evasive maneuver due to an NMAC event with an erratically maneuvering aircraft. The instructor evaded close to the terrain which triggered a terrain warning.

Narrative

After beginning our climbout of ZZZ en route VFR to ZZZ1 over the southeast end of the X Mountains I saw two planes appear on my ADSB-in. They were moving at a fast 168 kts. I was able to get them insight. They were behind and to the left of us nearby the mountains by City X. They were heading in a different direction than us. Once I had them in sight I realized it was only one plane with my ADSB reporting two- one as we were in a 172 climbing at 80kts. The other aircraft was red and I kept my eye on them as I was a safety pilot. They began turning erratically; it did not appear to be aerobatic activity. We were at the same altitude and I requested my co-pilot to remove his hood to keep an eye on the traffic. They turned towards us and away from us until heading directly towards us and maintaining our altitude with less than 100 feet of separation on the right side on the right side.I perceived the aircraft as a separation threat; as we attempted to maneuver out of his way he continually turned towards us and away in an inconsistent manner. The area is not an aerobatic area. It was unclear whether he had us in sight and his turns were sudden and at speeds upward of 160 KTS.We attempted a 90-degree turn to the left with a climb as the terrain warning system activated. The other aircraft followed us with the same turn and climb. At one point his turns were so erratic and he was less than 100 ft away; it looked as though he began to turn into us again and we attempted another 90 degree turn and climb to the left and attempted to stay close to the terrain; as it appeared he was also climbing when we would. He continued his climb over the mountains. And did not turn erratically again after passing us. I continued to monitor him on my iPad and with visual separation. He landed at ZZZ2.