In a fanfiction, they could be friends in an alternate universe where Harleen never met the Joker. They could work together to solve crimes in Gotham, with Harleen using her psychological knowledge to assist Batman in understanding the motives of the criminals. This would be a very different dynamic from the comics where their relationship is often adversarial because of her association with the Joker.
Another way to portray their relationship differently is to make it a romantic one. But not in the typical sense. It could start as a begrudging respect for each other. Harleen is drawn to Batman's sense of justice and Batman is intrigued by Harleen's complex mind. As they work together on a particularly difficult case, they start to develop feelings for each other, which is a far cry from their relationship in the original comics.
Their relationship could be portrayed as a more mentor - mentee relationship. Batman could see the potential in Harleen to be redeemed and takes her under his wing, teaching her how to use her skills for good instead of for the Joker.
One plot idea could be Harleen Quinzel trying to reform herself and seeking Batman's help in leaving her past as Harley Quinn behind. She might offer him information about the Joker's next move in exchange for protection and a chance at a new life.
It could be shown as a very strained relationship. For example, Wonder Woman might initially be very hostile towards Batman, seeing him as a threat to the order Superman has created. But as time goes on, she starts to see the value in his approach. There could be many tense confrontations between them that gradually turn into more respectful exchanges as they understand each other better.
Harleen Quinzel was a psychiatrist at Arkham Asylum. She became obsessed with the Joker during her attempts to treat him. His chaotic nature and charm gradually drew her in, leading her to fall in love with him. This ultimately led to her transformation into Harley Quinn, abandoning her former life and morals to be with the Joker.
The key element is her job at Arkham Asylum. It put her in the path of the Joker.
In fanfiction, their relationship could develop more quickly. Maybe baby robin shows an immediate trust in batman that takes longer in the comics. For example, in the original, robin might be more cautious at first, but in fanfiction, he could be instantly attached.
Their relationship could be portrayed as a mentor - mentee relationship in a different way. Instead of Orochimaru trying to use Naruto for his own gain, he could genuinely see Naruto's potential and guide him in the ways of some of the more esoteric jutsu. For example, Orochimaru could teach Naruto about the secrets of the snake contract without any ulterior motives. Another way is to show them as reluctant allies. Maybe there's a common enemy that is so powerful that they have no choice but to work together, and over time, they start to respect each other's abilities.
In some fanfictions, the Naruto - Sakura relationship is portrayed as a slow - burn romance. Naruto's continuous efforts to protect Sakura gradually make her fall for him. In contrast, in the original series, Sakura was mainly focused on Sasuke. Fanfiction gives an opportunity to change that narrative.
Their relationship could be portrayed as more of a brotherly bond. Goku could be like an older brother to Gohan, showing him the ropes of being a Saiyan and having fun adventures together.
Another way is to show a relationship full of regret. Anakin could be filled with remorse for how he treated Ahsoka in the past, but his dark side self still has the urge to do harm. Ahsoka, on the other hand, is conflicted between her love for the old Anakin and her need to oppose the current, dark Anakin.
Their relationship can be portrayed as more of a mentor - mentee relationship. Naruto, having learned from many great shinobi, could pass on his knowledge to male Haku. This would be different from the original as it flips the dynamic a bit.