Along the way, various tomes have vied for the spot as the "fourth core"; growing up in the 80s, I always felt like Deities & Demigods best fit that place, at least in terms of prominence within my young D&D-inspired imagination. But Unearthed Arcana, various iterations of a "PHB 2" or a Fiend Folio or MM2, an epic level book, not to mention setting bibles, have had their place.
Seconded! The first edition Deities & Demigods has a special place in my heart and has always been part of my "Core Four" Advanced Dungeons & Dragons tomes, melding my obsession with the game and my love for mythology. (Awesome avatar, by the way!)
And further, it opened up the game (for me, at least) in so many new directions... expanding ability scores beyond 18 (or beyond 18(00) for those who appreciate such obscurities), appendices expanding the scope and lore of the inner and outer planes, establishing gods for elves, goblins, dwarves, orcs and over a dozen more "nonhuman" humanoids.
(and then... sorta ninja'd by Umbran

)
Having said that, what would feel most "Fourthcore" (not a word) to me would be an Explorer's Tome (still workshopping the name)...
A book with guidelines and tailored sub-systems for two or three dozen fantasy genres... with maybe one or two new classes (or subclasses) and/or PC species for each. The new DM's Guide touches on this in the Creating Campaigns chapter and then gives us the Greyhawk setting for 24+ pages. But instead of pre-named, copywritten worlds, give me a variety of iconic genre settings, each with its own distinct flavor.
So you could have (among others) a steampunk chapter, Arthurian, fairy tale, a Greco-Roman setting, a high seas or swashbuckling setting, gothic horror and so forth; maybe a renaissance and a medieval couple of chapters, maybe a wasteland setting, a Western/fantasy mashup, and something akin to the Underdark? Maybe guidelines for magical school settings (I know Strixhaven is a thing), or even a generic world of competitive roaming monster-collectors.
Campaign setting books have their place, but I'd cherish a book of iconic campaign settings.