Not to take from that achievement away, though, I just don't think it fits in the context of this post!
In fact now I’ve confused myself, what would constitute doom running doom?
I think what is being discovered right now is whether DOOM itself is Turing complete, and if so, can you program DOOM inside of DOOM by making a computer inside of DOOM composed of DOOM monsters and map features that, when ran, recognizably played DOOM.
Sum up: creating a computer inside the game of DOOM and programming that computer to itself play DOOM would constitute DOOM running DOOM
In the genre of "an implementation of game X on an accidentally Turing-complete Y", it never hurts to bring up Tetris implemented in the Game of Life: https://codegolf.stackexchange.com/questions/11880/build-a-w...