There are several reasons. Firstly, fans are always looking for new and exciting storylines. Elena and Kol are both interesting characters with a lot of room for development in the love department. Secondly, it gives fans a chance to be creative and express their own ideas about love and relationships within the framework of these characters. They can play with different settings, time periods, and plot twists to make their own unique love story for Elena and Kol.
A great fanfiction could be 'Elena and Kol's Eternal Love'. This might show their relationship as something that endures through various trials and tribulations. It could start with their initial attraction, develop into a deep love, and then face obstacles like the disapproval of other characters or the forces of the supernatural world that they inhabit. Their love would be the central theme, with the story exploring how they overcome difficulties to be together.
There are quite a few fanfiction platforms where you can find stories about Kol and Elena's love. Archive of Our Own (AO3) is a great place to start. Just search for 'Kol and Elena' and you'll get a list of relevant fanfictions. Some might be rated high based on user reviews, which could be a sign of a good read.
People like to write about their love because it's a relationship full of conflict and passion. The initial animosity between them in the original story provides a great starting point for writers to explore how that could turn into love.
People write 'Elena Saves Klaus' fanfictions because it's an interesting twist on the usual relationships in the Vampire Diaries. It allows for exploring new dynamics between two major characters.
One possible plot could be that Kol, being the charming and somewhat dangerous vampire he is, initially catches Elena's eye in a moment of unexpected encounter. Maybe at a supernatural gathering. He starts to show her a different side of the vampire world, full of ancient secrets and forgotten powers. This new perspective intrigues Elena, and as they spend more time together, they develop feelings for each other despite the many obstacles like the existing relationships Elena has and the disapproval from other vampires and her friends.
One possible plot could be that Kol and Elena meet by chance in a place outside of Mystic Falls. They initially clash due to their different personalities and pasts. Kol is intrigued by Elena's tenacity and her kind heart despite all she has been through. As they are forced to spend more time together, perhaps because of a common threat, they start to understand each other better. Elena begins to see Kol not just as one of the bad guys but as someone with his own complex story. Their love could develop gradually as they share their secrets and fears, and finally, they realize they are in love.
Maybe the end would see them sacrificing their love for the greater good. They realize that their relationship is causing too much chaos in the supernatural world, and in order to save their friends and the town, they part ways. But they always carry the memories of their love with them, and in a way, it makes them stronger individuals in their respective worlds.
The relationship would shock Damon and Stefan. They have always been protective of Elena, and Kol has a reputation as a dangerous vampire. Damon might be especially angry and try to drive Kol away from Elena. Bonnie would also be concerned as she has seen Kol's past actions. However, Caroline might be more open - minded and could potentially try to help Elena understand her feelings for Kol. Klaus would be an interesting factor too. He might see Kol's relationship with Elena as a way to gain an advantage in his power games or he could be protective of his brother's new - found love.
They could develop a shared hobby. For instance, they both get into painting and use it as a way to express their emotions and connect on a deeper level. It would be unique because it shows a side of them that's not related to their supernatural natures.
In fanfiction, Kol and Elena's love often brings a unique dynamic. It can show Kol's more complex and passionate side. Their relationship might be explored in different settings, like in a modern AU where they meet in a different context than in the show. It allows for creative interpretations of their personalities and how they interact.