No, it's not common. The Naruto series is mainly about hope, friendship, and growth. Most fanfiction focuses on positive aspects like new adventures, relationships, or power - ups for the characters. The idea of Naruto being suicidal goes against the core of his character in the original work.
It's not a common type of fanfiction. The Naruto fandom generally enjoys exploring the more positive and action - packed side of the story. While there may be some out there that explore such dark themes, they are a small fraction compared to the vast amount of fanfiction that adheres to the spirit of the original series. Naruto is known for his determination and will to live, so this suicidal concept is quite against the norm.
I would say it's rather uncommon. Fanfiction writers usually like to expand on the existing positive traits of Naruto, such as his ability to inspire others, his never - give - up attitude, and his strong bonds with his friends. The idea of him being suicidal is so contrary to what he represents that not many writers would choose to go in that direction. There are far more fanfictions that focus on his heroic deeds, his training to become stronger, or his romantic relationships within the Naruto universe.