my cousin had a mod he applied to the game where you could somehow see 360 degrees around your character, he claimed to the end result of being able to more or less cheat in online multiplayer. It was very strange, and made the graphics appear somewhat shitty, which, for the time, they were absolutely not.
You can set the FOV (Field of View) in the console, he probably just modified that. We used to do something similar for Quake 2, set the view to about 120 and it was perfect for multiplayer. No weapon on screen and the ideal view.
Quake was kind of ugly, even in the context of 1996. It was a mass of brown textures wrapped around levels that were more linear and less inventive than what they created for Doom. The monsters weren't as memorable either.
Multiplayer is what saved Quake. For pure deathmatch, nothing beats it. The Quake 1 rocket launcher is the best deathmatch weapon ever created.
There was a fisheye renderer version of Quake which actually gave you a proper 360-degree field of view (rather than the usual rectangular projection).