Helicopter pilot reported a NMAC with another aircraft that appeared to be intentionally following them.

Narrative

On Date at approximately XA39 we departed from the Hospital located next door to ZZZ airport in a EMS helicopter.I made an initial radio call and heard and seen no traffic in the pattern. A crew member aboard called out traffic at 10 to 11 o'clock at a low altitude; we were showing around 1500 ft. and climbing. The traffic appeared to be very low only a few hundred ft. above the ground and appeared to be circling over some houses. I made another radio call stating our altitude; intended flight path; and intended landing location; and still heard no radio traffic reply.As we were continuing our climb up to 3500 ft. and our desired heading; the airplane then rolled out of his turn; starting climbing to match our altitude and then turned at our 9 o'clock to a heading that would be direct to the left side of the helicopter. Not knowing his intentions and the climb rate he was ascending; I stopped my climb and leveled out at 2500 ft. The airplane then passed directly underneath our helicopter at approximately 200 ft. below according to my onboard traffic avoidance system. I thought perhaps the airplane had not seen us and was not monitoring the frequency or did not have a radio. My crew member then stated the aircraft had passed so close as he flew under us that he could see his face and made eye contact with the pilot of the airplane.We then started our climb again and the airplane turned his course to match ours; and then leveled off directly under the helicopter with his altitudes varying from 100 ft. to 300 ft. below us. Not knowing his intentions I continued to climb to try to create more vertical separation but the airplane continued to change his altitude to maintain around 200 ft. below us. I then started to change my direction by 10 degrees to the right but then the airplane changed his direction to match still flying directly under the aircraft and still keeping around 200 to 300 ft. below us. I then turned to left; again trying to get this airplane out from under us; but again he changed direction to match. I made several calls on the radio but did not receive any replies.The airplane then turned slight to move to our 5:00 position still at the same altitudes and remained there for approximately 1 minute; then he changed course to move his position to our 7:00; all this time he was probably less than 1/2 mile in separation from us horizontally and during these position changes was now matching our altitude again. The aircraft then started a slow turn to the left and finally was moving away from us. The aircraft was now off our 10 o'clock position and flying very low in mountainous terrain. A couple of minutes later as I was on final; I heard him make a radio call that he was entering a left downwind for Runway XX at ZZZ1.My first thought as this situation happened was that the pilot didn't see us. It became clear though after the first near miss that the airplane pilot had a visual on us and apparently thought perhaps it would be funny to keep doing this. This whole occurrence lasted approximately 8 to 10 minutes. The pilot's actions were very dangerous and reckless. If I had not had two crew members to keep eyes on the airplane when he was not directly below me; or if I had not had a traffic avoidance system; this situation could have turned very bad. I could have been responding to a scene request and been landing in any of the fields below us in which case I could have been making altitude and heading changes that would have been unexpected by the airplane. I hope this pilot is contacted and at least educated of the dangers of his actions.