For mobile I don't think its a good idea, although I'd like them to work on granular refresh rate (adaptive sync like they're starting to do on discrete GPU). This way it could kick to a low refresh rate to conserve battery life (I'd love if it had a power miser mode where it disables animations and kicks the refresh low).
Honestly I'd rather they go minimalist. I don't need fancy animations, and most apps don't even need a quick refresh rate for anything (really quite a few apps could set it to refresh only when getting new data), it's just for frivolous animations and other junk that adds nothing to the actual experience (at least for me). But it would be nice to have options, which is why I'd like them to work on granular refresh control.
And going for higher refresh rates has benefits as well, like for VR headsets (which I think Oculus has said something like 85Hz is what the minimum for good VR with them aiming for I think 90 or so, but I believe they also use black frame insertion so the actual framerate of the content doesn't have to be that fast). Plus I would like them be able to offer fixed multiples of whatever content you're watching (so for 24fps stuff it'd have to be able to do 48 or 72).
Plus some devices I'd love where when you're on battery you could have it drop to a low refresh, but then when plugged in it'd go to a higher one (although again, I'd like to have some control).
Actually they've already done some with low refresh rates, that's part of why the LG G2 had such good battery life. But I'd like it to be more common and give the user and apps more control.
Also, right now part of the problem is the displays. With LCD they have to overdrive it quite a bit (would be curious about power figures, don't think I've ever seen that tested). And I think IPS still has issues just being overdriven that much, but pushing it up to 72 shouldn't be that much of a problem. I think OLED has more range (although not sure how far, but Oculus is pushing around 90 or so if I remember correctly). For desktop displays it shouldn't be a problem, but I doubt it's something they want to implement on most mobile displays, at least LCDs. Not sure if OLED has any power issues.
...and I hope it does. That 120hz thing is absolutely unnatural to me!
What 120Hz thing? I'm guessing you're talking about the TV functions? That's not quite the same thing. Also, it looks unnatural because you're used to low framerates for content.