> I can't remember if it came before or after Doom.
> But it must have been at about the same time.

Not quite. Ultima Underworld came out in 1992, way before Doom, which was released two years later.
But it was indeed about the same time as Wolfenstein 3D came out, which had, although the faster, technically the inferior engine.

I remember having read somewhere that it was a tech demo of Ultima Underworld which actually inspired John Carmack to write the 3D engine which would later become Wolfenstein 3D.
In the article was stated that he looked at the texture-mapped dungeons and then said something like "I can do that better."

Ultima Underworld, however, already had textured floors and ceilings, different height levels, angular walls, kind of a primitive light sourcing, even some gouraud-shaded 3d objects.
Moreover your character was able to jump, swim and climb.
All in all, Looking Glass (Blue Sky Productions at the time) did an amazing job for the time.