Their relationship becomes more possessive. In the Dark Side, Anakin's love for Padme could turn into an obsessive need to control her. He might see her as his possession rather than an equal partner.
One idea could be that Anakin's fall to the Dark Side causes him to try and make Padme a Sith as well. He might use his newfound power to manipulate her, believing that together they could rule the galaxy. For example, he could start by isolating her from her friends and allies in the Jedi Order and the Republic, making her more dependent on him.
Perhaps a fanfic that delves into the time when Anakin was on the cusp of turning to the dark side. Padme could sense the change in him and tries to prevent it with the help of Luke and Leia from the future who somehow travel back in time. This would create a complex web of relationships and a race against time to save Anakin from the dark side.
The fanfic could be set during the Clone Wars. Anakin's dark side starts to emerge due to the stress and manipulation. Padme notices the changes in him. Meanwhile, Luke and Leia are prophesied to be the ones who can save the galaxy from the dark side that Anakin is falling into. It could show how their fates are intertwined even before they are born, with Padme trying to protect the future of her children and Anakin.
Fanfics often explore how Anakin's fear of losing Padme drives his actions. He may see visions of Padme's death and in his desperation to prevent it, he makes choices that lead him further into the dark. His love for her becomes a double - edged sword. It gives him something to fight for, but also causes him to make morally wrong decisions. Writers may show how his relationship with Padme, which was once a source of light, gradually becomes corrupted as he descends into darkness.
In fanfiction, Anakin joining the Dark Side often has complex motives. It could be due to his fear of losing Padmé. The Dark Side might seem to offer him a way to save her, so he succumbs to its power.
In some fanfiction set in a more peaceful time, they might interact with a lot of tender moments. They could have picnics and long walks, just enjoying each other's company and having deep conversations about their future.
One possible reason for such a plot in fanfiction could be to explore the darker side of Anakin's character. His fall to the Dark Side in the official Star Wars lore was gradual, and in fanfiction, this could be an extreme manifestation of that. Maybe it shows how the power of the Dark Side can corrupt relationships as well.
A great fanfiction idea could be that Dark Anakin kidnaps Padme (not in a malicious way, but more out of his distorted sense of protecting her). He takes her to a remote planet where they are forced to confront their feelings for each other without the interference of the Jedi or the Senate. This would be a very intense and emotional story.
Obi - Wan Kenobi would be one. He's Anakin's master, and in the fanfiction, he might suspect something about their relationship and have to decide whether to support them or uphold the Jedi Code.