It's become a meme to even mention Half-Life 3 or Half-Life 2: Episode 3 or anything that involves a numerical value above '2' and the words “Half” and “Life”. However, there are actually some legitimate screenshots from the working title of Half-Life 2: Episode 4 that Arkane Studios had been working on many years ago.
The dedicated Half-Life website called Valve Time released a batch of new screenshots from the portfolio of Arkane Studios' environment artist Robert Wilinski, for a project they were working on between 2007 and 2008 for Valve, tentatively known as Half-Life 2: Episode 4. A detailed breakdown of how the assets were acquired and the details and secrets hidden within are covered in the Valve Time video below.
The game wasn't just another episode to what Valve had already established in the venerable Half-Life franchise, it was going to be a completely separate game with a brand new protagonist. Yes, they were going to put players in the shoes of someone else that wasn't Gordon Freeman.
Before freaking out, take note that this kind of spin-off of the Half-Life franchise was extremely popular during the first one of the original game, where players stepped into various roles of other characters in the first Half-Life, including Half-Life: Blue Shift, which was designed by Gearbox Software along with Half-Life: Opposing Forces. The former put players in the role of a security guard while the latter put players in the role of a military operative there with a squad to contain the Black Mesa fallout. Both games included new areas to explore, new characters to interact with and new weapons to use.
I imagine Arkane Studios' project would have been a similar spin-off to what Gearbox had done for Valve way back in the day.
According to the Valve Time video, the project was going to see players exploring the British town of Ravenholm between the events of Half-Life 2 and Half-Life Episode 2. The video points out a couple of Source Engine upgrades from Episode 2 that would confirm that the game would be using the latest version of Source before Valve began working on Source 2.
The screenshots don't reveal much but the video points to the locations possibly being associated with a prison basement, a sewer access or some secret location under a hospital.
Half-Life was known for its dark, sewer-style, corridor puzzles so it's not surprising that those same kind of locations would appear in the cancelled project.
What is strange about this whole thing is that recently DSO Gaming reported that some entries have been modified under a “Half Life 3” in the SteamDB. Some developers have been making some changes to the entry but the name itself could be a codename or alias for another top secret project. It's an interesting development nonetheless.