Situational Awareness; Distraction; Confusion
Conflict NMAC; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance
Human Factors; Airspace Structure; Procedure
I was conducting an SIC (Second in Command) checkride. We had just executed a missed approach out of ZZZ off the VOR-A. As we were climbing northwesterly on the missed; we got a TA for traffic at about our 2 to 3 o'clock. They were slightly above our altitude and began turning to the north as we leveled off and began heading more westerly. As we were visually searching for that traffic; we got another TA for traffic slightly below us and at 12 o'clock. We began to climb in order to climb above their altitude and continued our turn towards the west/south. I was unsure of their direction of flight and it ended up being southwesterly. We got to within 300 feet of vertical separation at the closest. Between the missed approach and the initial traffic; we were caught off guard by the second traffic alert. My suggestions is once traffic is in sight; ensure one pilot is monitoring that traffic; while the other monitors the instruments (and other traffic displays).
PC-12 pilot conducting a checkride reported a NMAC while executing a published missed approach. They received  a TCAS/TA for an undentified VFR aircraft; leveled off; then had to respond to a TCAS/RA for another unidentified VFR aircraft.