After takeoff from Belfast City Airport, shortly after the acceleration altitude and at a height of 1,350 ft, the autopilot was engaged. The aircraft continued to climb but pitched nose-down and then descended rapidly, activating both the “dont sink and “pull up TAWS (EGPWS) warnings. The commander disconnected the autopilot and recovered the aircraft into the climb from a height of 928 ft. The incorrect autopilot ‘altitude mode was active when the autopilot was engaged causing the aircraft to descend toward a target altitude of 0 ft. As a result of this event the operator has taken several safety actions including revisions to simulator training and amendments to the taxi checklist.