Just as a point of correction, Doomsday is not a Doom remake, it's an advanced source port. Freedoom (freedoom.sourceforge.net) is the closest thing to a remake, it's a project to create a free IWAD (the game data). It needs to be paired with an engine though, Doomsday is one, but there's also ZDoom (zdoom.org), PrBoom+ (prboom-plus.sourceforge.net) and my personal favorite, Eternity (doomworld.com/vb/eternity).