I don't know the exact name, but it is one of the Rex Begonias: hundreds of different varieties to choose from, but most of them very similar in requirements.
Generally, they like bright, indirect light, high humidity, fast drainage, thorough soaking when watered, then allowed to dry out a cm deep between soakings, and frequent, mild feedings with a low phosphate food.

Thank you, I bought two and the other one is more 'typical' of B. rex. I wasn't sure about this one, it looks more unusual.
I googled Begonia rex and found some images, and this one looks like it could be 'Indian Summer'
Thanks for your help with care also, I've never grown B. rex before so it's nice to know their requirements.

Could be--when I google "Begonia 'Indian Summer'" I see pictures of all sorts, including a few that look like this. Sigh...I remember when I used to get more or less what I asked for on Google... :/

Yes I get the same, but the first photos look more like this.
Also when I looked at sites selling it the photos and descriptions match this one too, so that's why I thought it was this one.

That's part of my problem: my computer software is so old, it doesn't allow me to get on many of those sites to check the actual pictures and descriptions. Google--and other search engines are even worse in my experience--often shows random images from sites that just contain those keywords, whether or not they are associated with a photo.

I find that with google too, it can be confusing and misleading.