Air ambulance helicopter pilot reported a near miss with another aircraft while transporting a patient and flying across a non-towered airport traffic pattern. The pilot did not see the conflict aircraft until crossing the airport and passing overhead the traffic; then continued the flight.

Narrative

Departing hospital with patient onboard. Due to winds a northerly departure course was favorable. The traffic at ZZZ was using [runway] XX. Broadcast my intentions on the CTAF. Heard an airplane on downwind and another taking off. Had visual on both aircraft. To avoid a conflict with the departing airplane I broadcast I would continue to climb but turn east and go around him before proceeding northbound. I heard two clicks as acknowledgement. This course would take me over the middle of the airport; but behind both aircraft. Upon crossing the centerline I heard an airplane call base; and another airplane call on downwind. I looked both ways along the downwind leg and then observed an airplane fly below me a few hundred feet eastbound. I broadcast that I was 2 miles north and had the airplane below me on downwind. I heard someone broadcast something to the effect 'good thing we were descending...' but no further communication. The flight proceeded without incident.ZZZ is known to be very busy with aircraft training. The hospital's proximity to the runway and the arrivals and departures to it are usually crossing the extended centerline. To prevent future similar occurrences I plan on simply circumnavigating the airport traffic area by a wide margin instead of picking my way across it.