Witchhazel is a pretty common shrub at garden centers and nurseries (i.e. real ones not walmart or home depot).
The most common kind are hybrids such as "Arnold Promise" (probably the most wellknown and arguably the best). They will bloom in late winter/early spring before the leaves are out, and also force well indoors (except that the darn things grow slowly so you can't be cutting TOO much for indoors or you will reduce your poor shrub to nothing!). They're hardy to about, what, USDA zone 5-6 [she says, not bothering to look it up but you should

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There is also another spring-blooming species, H. vernalis, that I hve no personal experience with; and a late-fall-blooming species H. virginiana that I really like as a native plant but it will never compete with Arnold Promise etc. The fall blooming thing still has its leaves on when it blooms although they will have yellowed, so unfortunately you can't really see the flowers unless you get right up close and stare at the branches. They smell nice though. Especially since usually by this time almost all other trees have lost their leaves for the winter and it's one last little 'first bloom of the year'
Witchhazels don't have any special requirements except a reasonable amount of water and at least part day sun. They are not fast growers.
I am not sure what 'bitterbush' is, although google sez there is a relatively obscure S FL shrub of the name... is it possible you mean "bittersweet"? If so, please read up on it a LOT before buying... most of what's sold is the asiatic bittersweet which is sort of like a northern version of kudzu and a serious invasive problem in many areas. The native american bittersweet is not generally considered suhc a problem, because it IS native, but I have to tell you that it can take over almost as badly as the asiatic can. This is NOT a small polite plant, it can pull down fences and damage trees with its weight, and once you've got it it's hard to get rid of. Mind you I do like the native one and have planted it myself but you have to make sure you know what you're in for
If you mean some other 'bitterbush' could you maybe find its Latin name so we can know what plant you are tlaking about?
Hope this helps,
Pat