So... is Omamori Himari finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: March 25, 2010
Manga finished: September 09, 2013
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
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
Yuuto Amakawa, an orphaned teenager with a severe cat allergy, discovers his life isn't so ordinary after all.
On his sixteenth birthday, he's attacked by a supernatural being and rescued by Himari, a cat-woman wielding a sword.
Himari reveals that Yuuto is descended from a family of demon slayers, and she's there to protect him now that his protective charm has weakened.
Now exposed to the supernatural world, Yuuto must navigate these new dangers and the intentions of various supernatural women drawn to him.
The story follows Yuuto as he adapts to his suddenly chaotic life and learns to deal with the consequences of his family's past.