According to the developers, at least a little bit of inspiration for Doom Eternal was pulled from the Christopher Nolan Dark Knight trilogy, but don't go in expecting a gravely-voiced caped crusader thwarting demonic crime. Instead, it all ties into an issue with the most recent entry in the series that proved mighty frustrating for id Software and the games community as a whole: Its name.
I think [Christopher] Nolan did it the best, and we thought about that. It wasn't 'Dark Knight 1, Dark Knight 2, Dark Knight 3.' Each one had its own name, but there was an arc there. They were all connected, as is this game as a sequel to [Doom] 2016.
This comes from Executive Producer Marty Stratton, who had a chat with IGN during the recent QuakeCon 2018. During the big show, Doom Eternal was given a rather lengthy gameplay demonstration, clocking in at 25 minutes of demon-slaying action. When asked about the game's name, Stratton confirmed that the team decided not to go with Doom 2 because of issues they had with the most recent Doom game.
Similar to what's being seen with the recently kinda-sorta rebooted God of War, it can be really frustrating to talk about a game when two entries in the series bear the same name. That's why, as Stratton admits, the entire id Software office had to refer to the most recent Doom as Doom 2016, because straight-up Doom already exists. This, obviously, leads to some problems. If you've tried to look up information about the original Doom on the internet anytime in the past two years, for instance, you likely had to fine-tune your search criteria because all of the biggest hits would be for the 2016 version of the game.
Stratton holds that he's happy they decided to call the 2016 entry simply Doom, but they realized they couldn't do the same thing with the sequel. So, Doom 2 was out of the question. As Stratton explains, the team decided they liked the way Christopher Nolan handled things with his Batman trilogy, giving each movie a name that distinguished it from the pack but also included it under a thematic umbrella.
Other game series have avoided these types of problems by taking a similar approach. The Devil May Cry reboot, for instance, went by DMC. A similar example is the new Tomb Raider series, which very closely mirrors the Nolan naming technique. They called the first game in the reboot Tomb Raider and likely ran into similar issues to Doom. The follow-up was known as Rise of the Tomb Raider, with Shadow of the Tomb Raider due out shortly.
As for Doom Eternal, we're still not sure when we'll have it in our hot little hands. Bethesda has shown off what appears to be a game in deep, deep development, so, hopefully, we'll be visiting hell on earth sometime within the next 12 months.