It's possible someone could pick up where Valve left off at some point-
Maybe a company like Bethesda?
I'd like that to be honest.
Not sure which developer would be good enough to continue Half-Life, but someone needs to buy the IP, since Valve doesn't give a rat's ass. I know, it won't happen, that would be like id Software selling DOOM. But just for the fun of speculation and theorizing... the real problem is indeed finding a good new home for the franchise.
Let's try for the heck of it (and by the way the list below doesn't take in consideration any devs or parent companies that wouldn't actually have the money to even buy the Half-Life IP to start with even if it was being sold):
1) Crytek? They'd make it look good, that's for sure. The game-play might be "ok", but the story and pacing would most likely be mediocre.
2) Infinity Ward? If anyone ever wanted Half-Life: Modern Warfare, they'd be the ones doing it.
3) Epic? Half-Life: Tournament anyone?
4) Monolith? Actually, we might be on to something here but... s'not like they actually made something good since the original F.E.A.R., still... food for thought.
5) Arkane Studios? To be honest I think they'd be the ones doing it [decent] justice out of all the candidates I can come up with. We need devs who understand the concept of story-driven, mostly-linear FPS gameplay.
6) Bethesda? I... don't think so. Unless we end up with Half-Life: Sandbox (honestly though, I don't want an open world Half-Life, I just want a sequel and a story well told, I don't mind linearity in that franchise).
7) Ubisoft? Half-Cry: Far-Life? Gordon and Alyx team up to activate radio towers t- ... no thanks.
8) id Software? Big interrogation point on that one. Let's wait and see how they can revive / rejuvenate their own IP first (sure, RAGE wasn't DOOM, and it wasn't a lot of things that a lot of people expected out of it, but at least we can say that they gave a try to story-telling in the FPS field).
9) 4A Games? Can't say much on that one, but I do know that Metro 2033 had good atmosphere (something Half-Life / H-L2 has, and still needs in a sequel).
10) Gearbox Software? Surprisingly (at first glance perhaps, but think about it) they might do something good, since at least Borderlands 2 had (in my opinion) great writing / dialogue and simply great gameplay overall. Granted, Borderlands has little in common with Half-Life, but conceptually I'm pretty sure that Gearbox would understand what a Half-Life sequel would need. And, obviously, let's not forget that Gearbox developed Blue Shift, Opposing Force and Decay for the PS2 version of H-L, so they actually do understand that franchise and its universe.
11) Creative Assembly? Half-Life: Isolation? Seriously though, still a better candidate than Infinity Ward.
12) MachineGames? Hey, Wolfenstein: The New Order was actually good.
13) DICE? Ermm... keep in mind that I just post a list of developers, as objectively as I can.
14) Irrational Games? System Shock 2, check. BioShock, check. BioShock Infinite, check. Heck, even SWAT 4? Nice. Alright, we have something interesting here.
I've surely forgotten a bunch. The devs above are the ones that work(ed) in the FPS genre specifically (either exclusively, or partially). But what about devs that haven't made a single FPS game before? BioWare? Blizzard (well ok, Overwatch, whatever)? That might also be something to think about.
Which dev(s) do you guys think would be best to take up the [very heavy] torch?