Both of them have the same quite glummy and mysterious atmosphere.
Mutual slow progression and quiet. Both are masterpieces in their own way.
Similarities--
~Same feeling
~The main characters both retain the same sense of hopelessness
~Beginning theme songs are a lot alike
~Both main charters are chasing someone
~Both main charters are booted from their 'sense of reality' and lose their position in society
~Both characters leave home in order to chase after the main antagonist
~Niether protagonist is a teenager
~Touch on romance but both main characters experience as failed relationships as they are both abandoned in a way due to their position in society and or loss of notoriety
Differences--
~One protagonist is male whilst the other is female
~Ergo Proxy is set in a future utopian society
~Ergo Proxy has supernatural elements as one of the
main characters is "god" and so are several of the minor characters
~One protagonist is a surgeon while the other is a dective
read more
Both shows are slow paced knotty psychological shows with heavy use of symbolism. Both require active viewing and possibly a rewatch to parse properly. As such both should appeal most to an adult audience that enjoys unravelling a story for themselves. Ergo proxy is more "on the nose" and confrontational but no less intricate than Monster, which is more methodical and lumbering, building tension slowly in a more reserved manner to an epic finale. If you had the patience to watch and parse one, chances are you'll like the other.
Monster is a psychological mystery thriller that is more grounded compared to Ergo Proxy. However both have realistic themes, philosophical elements, and intriguing mysterious characters, along with impressive world building.