Mine was one called "Zombie" on the Amiga/Spectrum/C64:
It was basically a game of Dawn of the Dead (in a shopping mall, you had to move trucks, escape on the helicopter, Hells Angels came at the end) but without any official endorsement.
I remember my terror as the zombies shambled past, noticed you and approached the screen and unleashed a flurry of undead blows at you. I didn't even dare go into the basement because it was dark in there....
What was the first computer game to terrify you?