Ok then, If your looking for something in deep shade I'd go
with begonias (any kind of fiberous begonia). They're slow to start for me - b/c they are small (I start from seed). BUT if you go to the nursery/Home Depot. you'll be able to pick up something larger. THey'll be more than impatiens $$ but they grow beautifully in a container. I'd go with single plants rather than a hanging basket - that way they can GROW into your container. They grow beautifully and full... These are NOT Dragon Wings, but they are happily growing in DEEP SHADE (NO SUN)
 I'm rather embarrassed showing this picture, but it shows the neglect... I threw three extra plants (these are dragon wings) into this 4" wide 'window box' on my shed. I neglected them terribly AND did nothing to them and they looked really nice...
 I just think this is tough to beat if you are into RED:
 BUT if you get more sun you'll have a larger selection (buying plants) if you go with impatiens. I'm not sure if they offer the larger blooming ones like blitz or envoy but, if you can get something like that they'll fill a container. The CLOSER you plant them the quicker they fill in and the taller they get. You don't want to plant them any closer than 6-8 inches apart though... this is a picture of a growing BAG I did - early...
this is later in the season:
 THIS is a picture of a smaller (dwarf) imp in a bag - it's nice, but not as vigorous as the larger blooming (envoy/blitz) imps.
 I grow them about 10" apart - early season:
This is a picture about Early August I'd say:
I hope I'm not confusing you:
Deep shade = begonias
containers = begonias look FANTASTIC
more sun = imps
better selection (colors)= imps
go with the larger blooming - not dwarfs (might be tough to do).
Imps = closer you plant the taller they get | 
