The second film that I want to recommend to you is Desu nôto, based on the cult anime Death Note (which is definitely superior to the film). Both works are based on the Takeshi Obata and Tsugumi Ôba manga of the same name, and are excellent gateways to this universe.
The film that I recommend is the Japanese version directed by Shusuke Kaneko, and released in 2006, although it is difficult to get it, it is well worth it. It should be noted that it is far from the version that is available on NETFLIX and that it loses much of the duel and the atmosphere created in the original anime.
In this case, the most relevant of the film are 2 points, which come from the original work and remain in the film, the transformation of a person by receiving a large dose of power, and the confrontation of 2 privileged minds, Light and "L". The development of the story is fantastic.

If you are looking for more action and less suspense, I can recommend Gantz. Based on the manga collection written and drawn by Hiroya Oku, it tells the story of Kei Kurono and Masaru Kato, two men who die in a train accident and become part of a kind of game where, together with other people who have recently died .
All of them are required to hunt and kill aliens armed with ultra futuristic weapons. Although in the manga and anime there are many explicit scenes (violent and sexual), these elements, for better or worse, do not appear much in the film.
In addition, the characters seek to know who they are and how they ended up there, and who the rest of the people around them are. Although it has good acting, it is an unpretentious science fiction film and from which nothing very deep should be expected. There is a very realistic animation version that can be seen on Netflix, which is called Gantz: O.

Closing the note, I recommend a film that I thought was excellent, Samurai X, also known as Rurôni Kenshin, also Japanese and directed by Keishi Ohtomo, the original story belonging to Nobuhiro Watsuki. The film was so successful that 2 sequels were produced, also based on the original manga.
Kenshin is a famous samurai known for his dexterity and coldness when it comes to murder. He knew how to fight hundreds of battles, claiming many other lives, and after finishing these confrontations, he vows not to kill again and decides to remain anonymous.

I hope you enjoy this selection, and it can serve as an approach to this culture that has so much to offer us.