I came on to make sure I, Claudius was gettiong some love as well! And of course Claudius The God, the second half of the story.
All the Natalie Haynes, the Pat Barkers (Silence Of The Girls, Women Of Troy and the third one due out this autumn), the Penelopiad by Attwood, Mary Beard, that others have mentioned - they are excellent.
I think Madeleine Miller's more a Division Two writer compared to those, but she's not bad. Stephen Fry... again, why bother when Haynes does it so much better.
I remember loving Women Of Rome by Moravia when I read it, but it was a good 30 years ago.
If you fancy some detective style novels, Lindsey Davis has a whole series set in ancient Roman times.
I agree if you want a change of civilisation, Neil Gaiman's Norse Myths* is very good.
*I was so delighted to get a signed copy at the time. It isn't quite the same now, though... Don't you wish people you admired would STOP turning out to be awful?