I'm surprised no one has mentioned the "Doomsday Engine," which is cross-platform and has the slickest modernization of Doom (I and II), Hexen, and Heretic that I've come across.

How much fidelity to the original to maintain is certainly a matter of taste, but imo they've done an excellent job of maintaining the essence while translating the basic feel and appearance to something slick and modern(ish): upscaled and better filtered textures, dynamic lighting, mouse look + strafing + jumping, UI for multiplayer match-making + game-playing, etc.

They've got videos and downloads here: https://dengine.net/