In a surprisingly low-key manner, Blizzard has effectively announced a new Diablo game, as they are currently seeking people to work on it.
The big question is whether this means a new expansion for Diablo III or if it will be a Diablo IV (or maybe they'll abandon the numeric naming conventions.)
Mechanically, I think they've done a fantastic job of tightening the game up over the course of Reaper of Souls. Seasons give you an incentive to go back and play more even if you're geared in best-in-slot sets and I really cannot overemphasize how much better the loot system is now than it was before Reaper of Souls.
A new game or expansion would hopefully keep up with this, but perhaps allow for more granularity in quality. Once at the level cap, the stats on a piece of gear are less important than legendary effects (particularly with class sets, though some sets are really unleashed by one particular item or another,) and so it might be nice to expand the "Ancient" legendary system to allow more of a sense of progress.
Storywise, there's plenty that's open. Given Malthael's hypocritical use of the Black Soulstone, all seven Lords of Hell are now unleashed. Likewise, Malthael himself may be dead, but as an Angel he should be able to return from the Crystal Arch (though hopefully the rest of the Angels are standing there, waiting to lock him up for what he did.)
Imperius has always kind of presented himself as a potential villain, given his disdain for humanity, though he was willing to help us fight Malthael, which suggests he might begrudgingly accept us.
However, I think the most interesting direction to take the Diablo series - assuming we don't find out about some other powerful entity outside of angels and demons - would be to delve into the history of the Nephalem. Humanity's ancestors (and the player characters in Diablo III) are effectively gods. Neither inherently good nor inherently evil (some, like Rathma, appear to have been inherently neutral,) the Nephalem could promise a far more complex story and environment for Diablo.
In terms of new features, I think the obvious thing with a new expansion would be a new class. The Crusader was a fantastic addition in Reaper of Souls (becoming my favorite class.) I'm not exactly sure what bases we don't have covered. I don't know much about Diablo II's Druid, but perhaps we could have a more nature-based class. The awesomeness that is Xul in Heroes of the Storm kind of makes me want a Necromancer, but that of course is already well-covered by the Witch Doctor, and on top of that, I don't know if they've ever repeated a class other than the Barbarian.
Anyway, we'll probably not hear too much about this for a while, possibly until this year's Blizzcon (which is of course several months away.)