In fanfiction, Naruto could be depressed after the war because he realizes that the world he fought for is still full of problems. The shinobi system might still be corrupt in some ways, and he may feel that all his efforts haven't really changed the fundamental issues. He might also be suffering from post - war stress disorder, constantly reliving the battles and the horrors he witnessed.
In some fanfictions, Naruto becomes more jaded after the war. He has seen so much death and destruction that he loses some of his initial idealism. He might be more cautious in his actions and less quick to trust.
In some fanfictions, Naruto and Hinata start a family right away. They enjoy a peaceful life in Konoha, with Naruto being a great Hokage and Hinata supporting him at home and also taking part in some village activities.
One common theme is rebuilding. After the war, the villages, especially Konoha, need to be rebuilt both physically and socially. Another theme is reconciliation, like Sasuke coming back to Konoha and being accepted again. Healing is also a theme, as the characters have to deal with the mental and physical scars of the war.
One possible reason could be to protect his family. After the war, Naruto might have made many enemies. By faking his death, he could draw the attention of these enemies away from his family, especially his wife and children. His enemies would think he's no longer a threat and stop targeting his loved ones.
One possible fanfiction could be where the depressed reader is transported into the Naruto world and meets Team 7. They start to find purpose in life as they train with Naruto, Sasuke and Sakura. The reader could initially be shy and withdrawn, but as they go on adventures with the team, they gradually open up and overcome their depression.
I recall a story where fem Naruto is depressed due to the constant rejection she faces from the village. In this fanfic, her depression is palpable as she tries to find her place. It's a really intense read as it also explores her relationships with other characters like Sasuke and Sakura, and how they either help or hinder her journey to heal.
Naruto could encounter wandering ninjas who have their own stories and reasons for not being part of a village. Some might be outcasts due to their beliefs or powers, and Naruto's interaction with them could lead to either conflict or new friendships. He may also come across some of the shinobi who fought on the opposite side during the war but now have different outlooks on life. Through these meetings, Naruto gets a broader perspective on the ninja world.
Naruto might become more empathetic towards Ino. After the war, he's seen so much pain and loss, and he could start to understand Ino's own struggles better. He may become more protective of her, seeing her as an important part of the village he fought so hard to save.
One common change is that he becomes more serious. He has faced the harsh realities of war, so he can't be as carefree as before. For example, he may not crack as many jokes during important meetings.