His personality could change quite a bit. The Horcrux's dark energy could make him more aggressive at times. He may also develop a sense of superiority due to the extra power, but also have moments of self - loathing when he realizes the source of that power is dark.
One consequence could be that his connection to Voldemort becomes even stronger. This might lead to him being able to sense Voldemort's actions and thoughts more clearly, but also make him more vulnerable to Voldemort's attacks on his mind. Since the Horcrux is a part of Voldemort's soul, it could also start to influence Harry's personality, making him more aggressive or moody at times.
One consequence could be that Harry feels an internal struggle. Since he has a part of Voldemort's soul in him, he might have to constantly fight against the dark impulses that come with it. For example, he may find himself having more aggressive thoughts or being more easily influenced by dark magic in certain situations.
If Harry Potter makes a Horcrux in fanfiction, it would be a major departure from his character. Horcruxes are dark magic, and Harry is firmly on the side of good. One consequence could be that he starts to be corrupted by the dark magic within the Horcrux. His friends might turn against him as they see him slipping into darkness. He could also lose his connection to the positive magic that has always defined him, like his ability to use the Patronus charm.
If Harry were to create a Horcrux in fanfiction, it would likely lead to a great corruption of his soul. Horcruxes are dark magic, and the act goes against all that Harry stands for in the original series. He might start to show signs of increased aggression, a detachment from his friends, and a growing attraction to the Dark Arts.
One possible consequence is that Harry becomes an even bigger target for Voldemort. Since a Horcrux is important to Voldemort's immortality, he'd be more determined to get to Harry. Also, Harry might start to experience some strange effects on his soul or magic due to being a Horcrux.
Well, it might enhance his abilities in some ways. Since a Horcrux contains a part of a powerful dark wizard's soul, absorbing it could potentially give Harry access to some dark magic knowledge or powers, though it would also likely be a dangerous and unstable influence on his magic.
One popular 'Harry absorbs Horcrux' fanfiction might be 'The Horcrux Within'. It delves into how Harry's character changes as he absorbs the Horcrux. The story explores his internal struggles and how it affects his relationships with his friends at Hogwarts. Another could be 'Absorbing the Darkness' which focuses on the magical implications of such an event. It shows how Harry has to learn to control the new power within him that comes with the absorbed Horcrux.
They're interesting because it's a unique twist on the original Harry Potter story. It adds a whole new layer of complexity to Harry's character. Instead of just fighting against Horcruxes, he has to deal with being a part of that dark magic in a way.
In some fanfictions, when Harry absorbs the Horcrux, it could lead to a moral dilemma within him. He has to struggle with the dark influence of the Horcrux while still trying to be the good wizard he is. This internal conflict can make for a really engaging story, with Harry constantly fighting his inner demons and trying to separate his own identity from that of Voldemort's.
Perhaps his father's old wand. It has a connection to his family and his heritage. Since it's a wand, it already has a certain amount of magical energy, so it could potentially be a suitable vessel for a Horcrux. And it would be a very interesting twist on the concept of a Horcrux as wands are not a typical choice.