So... is Monster finished or not?
YES, it is finished.
Why?
The anime finished after the manga:
Anime finished: September 28, 2005
Manga finished: December 20, 2001
MangaUpdates contains information about the anime adaptation and indicates that it is complete.
Watching Order (by release date)
Here you have all the anime relations we detected ordered by release date, so you can see the reason for this status in context and also use it as watching order.
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Synopsis
Dr. Tenma, a brilliant surgeon, makes a fateful decision to save a young boy's life over that of a more influential figure, a choice that costs him dearly.
Years later, the consequences of his past actions resurface, forcing him to confront the potential ramifications of his decision.
He embarks on a journey to right the wrongs linked to the person he saved, a quest driven by guilt and a desperate need to understand the true nature of his actions.