They are all each other's Constant by Leyla79

The island is underwater in Alt. It wasn't always underwater because Ben and his father were on it at some point. Because it's underwater, Jacob didn't seek out the candidates and they were free from the island to live their lives. This is what was supposed to happen (what happened happened). An alternate universe was created by Jacob, perhaps as a result of some catastrophic event that led to the island sinking, and he is seeking a candidate to prevent this from happening. Maybe - but the actual reason is probably immaterial. What is material is that he chose to bring these people to the island as potential candidates and to prove MIB wrong (THE candidate is likely the one who has been most resistant to corruption). Alt, in this case what happened happened, is a time loop and Jacob acted as a variable to break that loop.

Eloise knows that Jacob succeeded in creating an alternate time line. She was probably still on the island in the alt time line and may still have had some connection to Jacob. She had to leave the island because she was pregnant, or some other reason. Eloise has this same knowledge in the original time line as well. So Widmore has returned to the island in the original time line to either defeat MIB once and for all, or to ensure that the candidate takes his/her place to keep MIB contained. He brings Desmond because Desmond has the ability to travel between alternate universes. Desmond's goal is to seek out the original Losties so that he can flash back to the island to tell them of this alternate universe - of what was always meant to be. They will be given a choice to give up what they have in this life to go to the alt life. This can be done via death or by an electromagnetic event, much like the one that made Desmond special in the first place. The talk of love! in this episode is a throwback to the idea of destiny. These characters were always meant to end up in one another's lives in some way. Charlie is having visions of Claire during a near-death experience because his consciousness is altered and able at that point to recollect memories from the alternate universe. Daniel was able to remember his nuke idea through a dream - another altered state of conscoiusness. No one else has had similar recollections because they haven't yet been in such an altered state.

If the Losties choose the alt timeline via an electromagnetic event, their consciousness won't jump back and forth because they are each other's constants. When they make the decision to jump, they will remember one another in the alt and be grounded. Desmond's sacrifice will be either death on the island because Penny, his constant, won't be there to save him when the jumping becomes too intense. Or it will be to flash to this alt universe where he will have no memories of the life he had with her before. The Candidate will be forced to stay behind and live on the island fulfilling Jacob's role for eternity.

So I guess the question then is why offer this alternative up to the Losties at all? My guess so far is that the candidate will have to do something that will inevitably result in lots of death and he/she will be unwilling to do that without knowing first that in the end, everyone will be OK.
There is no reason why we can't see the candidate live on in the alt time line just because he stays behind. They are clearly two separate worlds existing in parallel at the same time. For e.g., if it's Jack, the effect of him being behind will be that the Jack in alt won't have the memories of the island.