Compaq has nothing to do with DOS. Compaq was a computer — and in 1983 it was a sensation, the first portable IBM-compatible machine that engineers could take home with a chunk of code worth tens of thousands of dollars. But a game with that name? That's murkier territory. If it exists at all, it's something from the very earliest era of home computing, when people were still figuring out what to actually do with these machines.
Little is known about the game itself. It's unclear what it was supposed to be — strategy, adventure, shooter? 1983 was a time when PC8088s with 8-bit processors and 64 KB of memory were mostly running utilities and simple educational programs. Graphics were non-existent, sound didn't exist. If Compaq-DOS was a game in the classical sense, it was probably some text adventure or the simplest arcade clone — something like Chess, tic-tac-toe, or a dumb ASCII shooter.
Playing on these machines was an ordeal. Black screen, green or orange text, every animation took forever. If you wanted to change the direction of your ship or character, you waited while it processed. Control was key by key, the mouse was science fiction. The fun was simply that something was running — and it was an achievement if it didn't crash.
Today such a thing means only speculation and nostalgia. Compaq-DOS practically never appears in archives — either it's some small obscure thing nobody documented, or it's confusion with something else. These oldest PC titles from the early eighties disappeared into the depths of floppy disks and forgotten catalogs, often without a clear trace of what they actually were.
It won't be in any browser and won't be on any retro archive servers. If you manage to dig it up somewhere, it'll be more of a curiosity than an experience — a taste of the time when personal computers were worse than toys. But that's what makes it interesting. An entire industry was born in these contraptions. And Compaq had its place in that.