 If you liked Monster |  ...then you might like Shouwa Genroku Rakugo Shinjuu |
These are two excellent character dramas. They both wonderfully convey the inner psyche of their characters, and go into great detail to explain the ins and outs of their thought process, growth, and interpersonal relations. They are also fantastic period pieces, really capturing the time and place of their settings. If you want to get heavily invested into some well written characters, both of these are great.