The games in this list are pretty bizarre and melancholic, because I'm pretty much of depressed and that can be the reason why I like these games. I won't make a long summary of them because my english is not that good, and these games are made to play, to feel them in your own. Lets get started.

The Rule of Rose

This is a PS2 game and the graphics are not that much, but is the "history" that we are interested. All in Rule of Rose is symbolic, because it's all happening in Jennifer's mind, is all mixed up, bizarre, and Jennifer objective is to remember things, in the game she has help of Brown, a dog that she find suspended in a rope, Jennifer can also carry weapons but the combat system is pretty bad, and in most cases the best choice is to run. In the end you remember your history, but I'll not give "spoilers". Try it for yourself, especially if you like those "in mind" games like me.
Rule of Rose has a intro that gives me shivers:

Downfall

This is the most fucked up game, serious, it's so fucking bizarre. The plot is around a hotel named "Quiet Heaven" that Joe, the protagonist, took his wife, Ivy, trying to refresh their relationship. Joe and Ivy had a fight in their room, despite of the recepcionist asked them to not wake the woman in the next room. Then things get a pretty bizarre, Ivy is gone, and the recepcionist told Joe that she is with the woman in the room next to them, Sophie is the name of the woman, when Joe is able to enter Sophie's room, he found a woman with a mask sitting in a coach and the room is full of mirros, Sophie asked Joe to kill his 4 memories, she included, to be able to get Ivy back, and the rest of game is the search of Joe to kill these memories.
The game has so much history than this, but the important to me, that make this game a masterpiece, is the fact that Joe is schizophrenic, in the end of the game you can conclude that and if you also played The CatLady you already knew that. Beyond Joe being mental ill, Ivy has signs of depression, and every Sophie's memory is a mental state of Ivy, the Sophie's memories were increasingly fat. And all the characters in the game represent something of Joe, like his perfect woman, the part of Ivy that he fall in love, his asshole part. You all have to play to feel what I felt, my words are not capable of expressing it.

The CatLady

This is the #1, maybe because I played it before watching videos or maybe I just identify with Susan. Susan, the protagonist, is depressive and starts the game trying to kill herself, after that she wake in a different world and explore until she finds the "Queen of the Maggots", an old woman that tells Susan she is now immortal and have to kill 5 parasites to be able to die. Susan wakes in the hospital and further is kidnapped and killed by the doctor, one of the parasites, she revives in the doctor's bunker and kill him. Returning home Susan meets Mitzi, the complete opposite of her, and after a while they become friends.
Different of Downfall, Susan isn't mental ill, the most bizarre moments is when she dies and find herself in a intermediate world. The history is pretty good and more than that, if you have depression is a game that you have to play over and over again, because it isn't cheap self-help, it's a badass game that, depending on the final you achieve, will make you feel better.

P.S.: The soundtrack of all games is marvelous, this is something that I take in high consideration, I'm a music lover.