I am very surprised that anise (Pimpinella anisum) would root in water. Is it possible you bought Anis hyssop (Agastache anethiodorum)? That's a member of the mint family and would be much more likely to behave as you describe. I have never grown it so can't advise, but if we can establish the correct identity someone here will help you.

Aha! You are right. It doesn't look like anise at all - it doesn't even look like anise hyssop! I think it is a Thai basil, maybe Queen of Siam.

I just accepted the name given by the market vendor, without question, and I have never grown anything anise-y. Thanks for pointing me in the right direction.

So... do you think I can plant them in pots and winter them over in a cool room?

Here is a link that might be useful: Thai Basil profile at Dave's Garden