I really like it. It has a hedge-like growth habit, and those really pretty leaves. As you can see, the flowers are tiny and would be completely insignificant, except that they float like a cloud over the whole plant. This plant is growing in north central Florida.
SOLVED: I\'m growing to love this plant
Definitely Persicaria, maybe 'Red Dragon'? I had it in a large container for a few years where it seemed perfectly hardy in the Atlanta area. Alas it died after the fifth winter. If I could find it again I would try it in the ground as it did not seed invasive like its many other cousins with their millions of seeds. Check out Painter's Palette.
Thank you! That's definitely it. I'm impressed that it lived for you for so long in the Atlanta area! In any event, a really pretty plant and, as you say, doesn't seem to be invasive. Thanks to both of you.