So... is Honoo no Alpenrose: Judy & Randy finished or not?
NO, it is not finished.
Why?
The anime finished before the manga:
Anime finished: October 05, 1985
Manga finished: May 01, 1986
MangaUpdates contains information about the anime adaptation and indicates that it is incomplete.
Watching Order (by release date)
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Synopsis
Honoo no Alpenrose: Judy & Randy tells the story of Randy, a young man in Switzerland, who discovers a girl, Judy, the sole survivor of a plane crash.
Judy suffers from amnesia, so Randy takes her in and raises her. They grow up together in the beautiful, yet turbulent, Swiss countryside.
Years later, Judy, now a teenager, feels compelled to uncover her forgotten past. Her only clue is a haunting melody she calls "Alpine Rose."
Driven by this mysterious song, Judy and Randy embark on a quest to discover her true identity, navigating a world increasingly gripped by the chaos of war.
Their journey will test their bond as they face numerous challenges, relying on each other for strength and support amidst the turmoil.