Either Barbarian (Psygnosis) or SDI (Cinemaware).
The computer was an Amiga 500 and the year was 1988.
Barbarian was included with the Amiga 500 that my father bought. He also purchased SDI at the same time as the Amiga. I can't recall which game I played first.
A bit off topic..
I remember the first that I saw a screenshot of an Amiga game. It was on the backside of a "Borrowed time" box. I was amazed by the graphics because it looked so much better than the IBM PC CGA version of "Borrowed time". But a couple of years later when I had an opportunity to play the Amiga version, of that game, and I thought it look quite bad.
I also remember the first the I saw an Amiga game in real life. It was the intro to "Defender of the Crown" played on an Amiga 2000 in a computer store. The golden logo looked so much better than anything I had seen on a computer at that time.
