The glasshouse flower if you need bloom almost any month of the year the Poppy Anemones should be more commonly grown as these are also so useful for cut flowers. Anemone coronaria is the original wild species native to the Mediterranean region and this makes it more tender than most other species many of which are fromcool woodlands.
Cross-breeding has improved this tuberous relative of the buttercup and there are two well know lines the St Brigid and De Caen both often grown in outdoor gardens. Sadly then, except in the mildest warmest sites these beauties are very short lived.
Under cover though these survive longer, more strongly and produce wonderful unsullied blooms, perfect for cutting, whereas outdoors the weather spots and ruins them. So good are these another name is the Florists Anemone. Anyway indoors or out even if the tubers survive their flower display gets less impressive within years and it is better to replace the tuberous rootstocks after three or so. (Or better still replace a third every year.) These are so inexpensive this is no problem, especially when you remember what most cut flowers cost to buy!
Now in a cold-frame, cloche, warm site these bloom in April or May. However in a frost free greenhouse you can plant these in batches from September till April to get flowering as early as February and then on through much of the year. And with warmth as well, and with a tad of extra artificial light, you can even get blooms just about every month of the year.
Apart from the little extra complication if you want to force Poppy Anemones these are actually pretty easy. (Though with bought-in rootstocks soak the funny little tubers over-night before planting them.) They must have good drainage and full sun and prefer a humus-rich cool but never wet or waterlogged compost. Deep containers suit them more than shallow pots and these are happiest in a deep greenhouse border where they under-plant well with lilies and other bulbs for cutting.
You could even breed your own varieties as these are relatively quick and simple from seed. However with so many shades of blues, whites and reds, singles, and doubles there are many excellent named sorts to grow first.
A final note, Poppy Anemones are excellent cut flowers, long stemmed they will stand a week or more AND much the longer if the cut end is put under water immediately after picking.