I just finished my first run on Normal difficulty and I see there are 2.5 higher levels of which I can select 1.5. I see how to move my current character to Elite but am not sure of the point. Do most people who have finished BoC do something thereafter with their Normal toon in Elite, or do people usually start a new alt? How can one decide?
I think most people move onto Elite, and the into Ultimate difficulty. Normal is just the warm up…
That is typical for this style of game. You beat the easy difficulty(normal/veteran), then you do the next one (elite) and the next one(ultimate)…then you farm and/or start an alt.
The “point” of moving up is the challenge as the story is the same. You do carry over all gear/skill and reputation, just the quests reset and some shrines disappear.
You could replay some (high level) content on veteran mode to get a grasp of what awaits you on elite. If you start a new toon on the first difficulty (either normal or veteran) or move on to the next difficulty is entirely up to you.
There’s always the possibility to switch back to a lower difficulty. You won’t lose the progress in the actual difficulty. And your rep will remain the same too, so if you get the next rank in faction rep in Elite you can go back to normal/vet and do the related quest there. Or if you discover a shrine in a higher difficulty you missed in normal you can always go back and activate it. Or if you just want to get stronger in a higher difficulty and then go back to do the harder quests in the skeleton key protected areas.
" You do carry over all gear/skill and reputation, just the quests reset and some shrines disappear."
That’s the point I’m groping for, I think. So if I now elect Veteran or Elite difficulty, I still have a level 60 toon and a collection of tools, but now all the stuff I did on Normal is to do over on the higher difficulty level?
And if I start a new alt, can I immediately restart the game on Elite difficulty with my new alt, or would that be futile?
No difficulty unlocks are per character. If you could play elite on fresh toon you would get absolutely shit on 
And just to clarifiy, Normal and Veteran count as the same difficulty mode. So everything you do in Normal is also done in Veteran and vice versa. New start is only done in Elite and then Ultimate.
It is also impossible to reach full 50 devotion points without playing Elite and Ultimate, because there are 28 Shrines in Normal and 17 in Elite, and (I think) 8 in Ultimate.
Additionally there are additional attribute and mastery points, which are quest rewards. So you need to play through Elite and Ultimate to max out everything.
OK, I think I got it. Appreciate the help!
But there are small differences too, there are less shrines (and the ones that are there are not all at the same location) in Elite like in Normal/Vet and you also have the opportunity to choose a different faction (though I’d do this in Ultimate (or) after having maxed rep and bought all items you want, because rep then changes and gets carried over in all difficulties).
probably the biggest reason to switch to Elite and then Ultimate is that Monster Levels cap, and the best legendaries and epics simply can’t drop in Normal.
And you don’t ‘move’ the char to Elite, you select which mode to play in, and you can toggle back and forth as you wish.
Just to clarify, if you choice Death’s vigil in Normal/Veteran you can change to the Order on Elite or Ultimate. This allows you access to items by from both factions. But the key point he the above poster makes is “bough all the items you want” as if you change factions in Elite/ultimate you will loose access to those items in Normal/Vet as your rep with the old faction will reset back to zero. If you stay with the same faction, your rep continues where you left off… So if you go that route, it is a good idea to buy items (mostly augments) and stock pile them in your stash. Then you’ll have enough of either one. Does require some work to get the needed rep for one then switch to another one… If you keep with one faction, you probably won’t need to grind much at all…
The saved progress in a previous difficult can be useful. There is a black smith in Tyrant’s Hold that allows you to trade in brains/hearts/blood for legendary items. You won’t find him there until Elite. But the black smith that lets you convert your bloods/hearts/brains to whatever one you want is not found until the necropolis. So if you’re in elite (and you have a bunch of these rare mats) you can jump back into normal and change then out at the blacksmith in necropolis then then jump back into Elite and run to Tyrant’s Hold BS. Also faction venders will sell you the same items no mater what difficulty you are in. So if you’re not to Homestead yet in Elite, but want some homestead faction gear you can jump back into normal, get your gear and then go back and continue with Elite.