Gap Dong (2014)

Both dramas have incredibly intelligent psychopathic serial killers (especially Gap Dong).
If you enjoy those kind of stories you will definitely like both of these dramas.
While Hello Monster has a really complex story with incredibly diverse characters, Gap Dong's story is way darker and focuses more on the nature and the perspective of the serial killer and his actions.

-----(minor spoilers following)-----

In my opinion the actors for Hello Monster were all incredibly well cast which helped to improve the story and made it easier to portray the characters. In contrary for Gap Dong Lee Joon did an amazing job on portraying a coldhearted psychopathic killer, while in my opinion the other main roles were poorly cast/ badly portrayed, which didn't really help the storyline...

Both the movie and the drama are based on the same true events. So there could be a lot of similarities among them.

A beautiful killer uses the system to hide in plain sight.
VIP has the international aspect where GapDong small town based.

I'll just show the summary but I'd say that it's similar with the questions of who could have done it then and are they starting back up or did they never stop, or is it possible that they're copycat kills.

"Moo Yeom is a detective. 20 years ago, he was a Buddhist monk, but he got into trouble with a man named Gap Dong. To cut ties with him and cut off whatever karma would arise from his involvement with Gap Dong, Moo Yeom became a detective. After the statute of limitations expired, Gap Dong appears once again. Moo Yeom sets out to catch him once and for all."

Both are psychological thrillers exploring someone else living inside you.

In both there is a suspicious cast of characters AND a whole lot of dark secrets

Both are about chasing a serial killer and corrupt higher ups.
Worth watching drama.
Chilling thrilling.

Both are about chasing a serial killer and corrupt higher ups.
Worth watching drama.
Chilling thrilling.

[Spoiler] The eighth episode of this drama and drama Gap Dongi have the same theme, about a serial killer disappeared in the late nineties after his 14 crime, and now after 17 years of the killer disappearance, crimes appeared again and appallingly!

Both dramas are about serial killers and copycats. Both main lead detectives affected by serial killer in past.

Both dramas has:
-smart characters as the main ones.
-killers.
-detectives who are involved in the case.
-characters with a deep story life.

Both are crime dramas produced by tVN, and both are inspired by the movie, "Memories of Murder".

For the serial murder mystery angle. Both shows showcase a cop trying to protect a witness/near victim of the serial killer, with a slight romance or perhaps pity for the victim angle, as they chase around the countryside.

The serial killer character is very similar.
Both are K-Drama.

Both feature a male lead with split personality (Kill Me Heal Me has several personalities while Gap Dong only has two) and an unpredictable plot based around mystery. However Kill Me Heal Me is for the most part more light-hearted and comedic than the rather heavy storyline of Gap Dong.