This was popular with Symbian S40 and S60 devices, completely dropped off in the smartphone era along with the FM radio.
Do you remember you could "turn off" your device while having your clock and alarm running? Basically the device's operating system was put into a "deep sleep" mode when you turned it off (not really though) and at the same time you could get the alarm running; no notifications, no ringings, just the clock and alarm.
I know I can get a similar effect by putting the Plane Mode on but on the old Symbians the device stopped most of the undergoing services and used the RTC (real time clock) which usually runs at hardware mode.
The benefit is mostly you can save some battery and reduce EM waves influence while sleeping.
Most feature phones implement it...so why not on Android?
(edit: I had to delete and rewrite the post as the site didn't like spaces and general syntax the first time :D )
Tl;dr : It could be a very welcome thing!
Sorry to be late but I don't always keep eyes on notifications :)
I used to have 4 different Lumias, one is still my backup device (on N7+ now) and none had it, last Nokia Symbian I had that used this feature was E63.
It's funny while I point out the feature that other people tells me that modern smartphone are designed to be always on, which in fact is and has always been (there has never been a mechanical switch, was it?) .
What happens is that, at least on Symbian, " switching off" paused the notifications, network services, UI services, user settings but could not turn off (like you took the battery out); I can only think that Android does the same, and basically if you turn battery saving and Plane mode and put device into standby you get almost the same result.
BUT, you get this with THREE steps ! Symbian could do that in ONE step (long press power button) .
Such feature can be really welcome to people needing to save batteries for a bit longer, when no power outlet is immediately available, and can spare some electromagnetic interference while sleeping (better than any sleeping manager IMO).
And, why not, nostalgia effect (marketing, you know?).
I wish some HMD engineer could read it and assess the idea of getting this working at last.