There's a fanfiction called 'Left Behind'. Here, Edward leaves Bella thinking he's losing his humanity and becoming a danger to her. Bella refuses to accept this and tries to bring him back to his senses. She delves deep into the history of vampires and their emotions to find a solution. It has a very emotional and engaging plot.
The fanfiction 'Forsaken Love' is quite unique. Edward leaves Bella after having a premonition of her death if they stay together. However, Bella doesn't believe in premonitions and she sets out on a journey to find Edward and convince him that they can change their fate. Along the way, she discovers new powers within herself.
In such a fanfiction, Bella might experience a whole new life. She could gain new powers and abilities as a vampire. For example, she may have enhanced strength and speed. She might also face new challenges in the vampire world, like dealing with other vampire clans or trying to control her new thirst for blood.
In some fanfictions, Edward leaves Bella because he's cursed. He doesn't want Bella to be affected by the curse. Bella, being in love with him, tries to find a way to break the curse. She travels far and wide, meets different supernatural beings, and in the process, she also becomes more powerful. When she finally finds a way to break the curse, Edward realizes how much she truly loves him.
One possible plot could be that Edward, perhaps influenced by some dark magic or a misunderstanding, decides he has more in common with Harry. Maybe he sees Harry's courage and selflessness and feels drawn to that world. He leaves Bella heartbroken, and Bella then has to find a way to move on while Edward starts a new life in the wizarding world with Harry as his new friend or more.
Bella's character might become more confident. Without Edward, she has to rely on herself, which forces her to grow. As a vampire, she'll be more physically capable, and this could translate into mental strength as well.
There's also 'Edward's Redemption'. In this fanfiction, Edward has to face some tough consequences for his actions in the original story, and it shows how he tries to make it up to Bella. It delves deep into his character and his love for her. This one is quite popular among fans who like more complex storylines.
Well, there's 'Midnight Sun Rewritten'. It's a fanfiction that takes the concept of 'Midnight Sun' and gives it a new spin. It's really interesting to see how the author plays with the original ideas. Then there's 'Bella's New Beginning' which changes the starting point of Bella's journey into the vampire world. There are also fanfictions like 'Edward's Secret' that reveals some hidden aspects of Edward's character in relation to Bella.
One fanfiction has them honeymooning in a desert oasis. It's not what you'd typically expect for them. The oasis has beautiful, clear pools of water surrounded by palm trees. They have to deal with the heat of the desert during the day and the cool nights. It adds a different kind of dynamic to their relationship as they face the challenges and joys of this unique setting.
I've read one where Bella discovers a new power as a vampire that allows her to communicate with Jacob telepathically. This completely changes the dynamic between them as they can now share their innermost thoughts and feelings without having to be in close proximity. It also makes their relationship more complex as they have to deal with this new form of connection.
One possible reason could be that Bella discovers a side of Rosalie that she feels more connected to on an emotional level. Maybe Rosalie shows her a different perspective on the vampire world that Edward had been hiding from her. Another aspect could be that Bella has grown tired of Edward's over - protective nature and sees Rosalie as a more independent and free - spirited vampire to be around.
You could make their relationship more unique by having them communicate in a different way. For example, instead of just talking, they can have a mental connection from the start that only they can understand. This would make their bond more special and exclusive.