Anime

I haven't delved too deep into the weirdness of anime, but probably the most confused I ever felt watching an anime was the ending of Neon Genesis Evangelion. I originally watched the series when I was in high school and couldn't really follow what was happening in the final couple episodes. I distinctly remember thinking to myself that there is probably some deeper meaning here that I am just too dumb to understand...

...and I still think that. I am sure if I really wanted to read explainer articles or watch more of the many Eva-adjacent things it would make more sense, but I am happy to just not fully understand it. I am content with appreciating it as an influential piece of art within the genre and move on.

Furi Kuri and Kuuchuu Buranko.

Furi Kuri is a favorite for some but a lot of the images didn't make a lot of sense to me. I like TheCynicClinic's video essay on the series though.

Kuuchuu Buranko is also really weird. The narrative is easier to follow but it uses a lot of weird animation techniques. You can probably describe it to be visually psychedelic. It introduced me to Denki Groove though.

Cat Soup, I didn't understand enough of what was going on there to even describe it.

Out of context mildly NSFW screengrab (it doesn't make any more sense even in-context)

Reincarnated as a vending machine was absurdly fun.

Perfect blue seemed to be preoccupied with pants pissing

Serial Experiments Lain. Assuming I remember it correctly. Or understood to begin with. It’s been a while.

If anyone's curious what the one from OP's picture is it's called Angel's Egg. Seen it a few times now, very slow, dark and quiet. I love it.

I guess probably Kaiba (and it's also one of my all-time favorites).

It's odd though, because the settings and art design and such are very, very weird, but the storylines themselves - both the individual episodes and overarching plot - are actually fairly straightforward. It deals with some fairly serious themes - love, loss, loyalty, hope, betrayal, redemption and so on - and it does it very well really. It just also does it in very weird settings with very weird characters.

For that matter, FLCL qualifies in the same way. I'm not sure why I didn't think of it right off - I guess I stopped noticing how weird it is somewhere along the way, because it's just... FLCL. It is what it is. But it is very weird. In the same way as Kaiba though, behind the very weird details is a fairly straightforward story. And in a way, it's even simpler - where Kaiba deals with some relatively broad and complex societal issues, FLCL really just deals with a boy coming to terms with growing up in a bleak nowhere town, and starting to sort out how to deal with the opposite sex.

And as long as I'm here, I want to mention Ergo Proxy, which is exactly the opposite type of weird. Aside from a highly stylized science fiction setting, it's really pretty straightforward from moment to moment. It's odd, but no more odd than should be expected from the setting. But all the while, behind the current things going on, there's this background story that's bizarre to the point of near-incomprehensibility.

Utena

Ooh interesting question and I'm sure I'll need some time to even remember some of them.

The ghibli movie with tanuki is what comes to mind. Japanese traditional lore consistent with transformations, tricking humans, and big and huge nuts - or sth. I don't quite remember it well.

Urusei Yatsura of course... Hen to hen o atsumete, motto hen ni shimashou!

Happy Sugar Life. The name did not resemble the anime at all.

Katte Ni Kaizou. These piss bottles...man.

How tf has nobody said Made in Abyss yet?

This shit has downright freaky abstractions who wanna use magic soul currency to trade you for body parts to get their desires satisfied!