It's an awkward, unpredictable show. Sometimes it's wonderful. Sometimes it's hard to watch, for various reasons. Sometimes it'll annihilate you, but I've seen some vicious hatred for the final episode and the fan backlash made Jun Maeda delete his Twitter account.
Jun Maeda is a huge name, by the way. He's co-founder of the Key visual novel brand, inventing the "nakige" genre that made you cry. (This is different from "utsuge", which is merely depressing.) He worked on Kanon, Air, Clannad, Angel Beats! and Charlotte. (Also Little Busters!, which I dropped on finding that its first episode seemed to be about fights and baseball. I'm sure that's unrepresentative of the show as a whole, but what the hell.)
Youta is a nice lad who's studying for his exams and occasionally playing basketball with his friend. The god Odin (aka. Satou Hina) is a small pink-haired girl (estimated age: ten?) with a ton of attitude and the ability to predict the future.
She decides that she's going to live with Youta and change his life in the thirty days they have left until the end of the world. No, he doesn't get a choice in the matter.
Hina/Odin is glorious. Everything she says and does is entertainment gold. She orders everyone around, despite being about a foot shorter than them, and has an endless supply of overreaction faces. She thinks she's magnificent, infallible and sexy. (No one else agrees with the latter, but it still leads to a brief idiot comedy sequence in ep.6 that just about murdered me.)
Youta is a lovely chap, as are all his friends. It's a warm, happy show... until it really, really isn't. (More on that later.)
AWKWARD STUFF THAT I PERSONALLY HAD TROUBLE WITH
In the early comedy episodes, Hina's do-gooding boils down to making Youta do something cringeworthy. I struggled with this. In ep.3, he claims to be a middle-aged ramen restaurant consultant and his sister's uncle. He overacts something rotten. I realise that it's the silly lies that make these situations so explosive and hence (perhaps) funny, but I couldn't buy lines like this. "You see, I've been doing research into magic and I accidentally ended up inventing a phone that connects to the afterlife."
(Despite appearances, this is a real-world show. There's no supernatural.)
The show's charming and funny, yes. At its best, it's one of the most entertaining shows of the year. I did, though, struggle a bit. Then, though...
THE SHOW GETS HEAVY
Wow, does it.
There's an explanation for how Hina turned into Odin, as per the title. There are some powerful people who make a decision that'll make you want them dead. The last three episodes in particular jump into a completely different genre. Youta makes some painful moves because he's being a typically optimistic anime protagonist and gets corrected mercilessly by a therapist. He's wrong and she's right. This is a bleakly realistic situation and you can't just fix it with positivity.
...and then comes the final episode, which makes some decisions that some viewers hated, hated, hated. Personally, I accepted it. Logos Syndrome isn't real. It can behave however Jun Maeda likes. It's not taken so far as to be only explicable as magic, but Maeda has form with that too. Speaking as a viewer, I'd already been watching in magical realist mode as soon as I saw his name attached to the project. The therapist's decisions are comprehensible (although debatable) and the emotions feel true. The show's exploring difficult ground you won't often see broached and certainly not again like this.
SHOULD YOU WATCH THIS?
Maybe. I seem to like it a lot more than most, but I don't imagine I'll rewatch it. I prefer it to Charlotte, though, which was just as rough a ride but less charming and more awkwardly paced. I'll definitely remember it, though. It's one of those shows that'll still be with me in ten or twenty years time and my memories won't be negative ones. I admire the show and I'll definitely be in line to watch pretty much anything Jun Maeda writes.
Personally, I bought the ending... but not all viewers did. Which is putting it mildly.