Tuberous begonias tend to lose all their stems over winter and have very showy flowers over the summer. What you might have is a begonia coming from a rhizome instead. Many of that sort have attractive leaves and are usually nearly 'evergreen'.

If you treat it as you would a succulent: protect it from frost, water sparingly over the winter - and refresh the mix when it starts into fresh growth.

If you want to propagate it - take a piece from the actively growing tip of the rhizome (about two inches) and plant it as you would for the main plant. It's better to use a small pot than a big one. They resent being overpotted.

Another way which can work is to take a single leaf and stem and put it in water until it begins to form a root system.

I don'y have the space for much seed-raising so this is 'second-hand' info: mix the begonia seed with fine sand and sow on top of a tray of damp seed medium. Do not cover it. Do this in winter or early spring and hold the temperature between 61-66F. Light without heat and dryness - so that's probably gro-tubes or similar. A humid environment helps prevent the young seedlings from drying out. Probably sow thinly - to make it easier later on to prick out the young plants. I would suspect they would be up by early summer.

It could be worth trying the germination rates for seed that is not fully mature with seed that is. Sometimes the slightly immature seed in other species will germinate more readily than the fully ripe which has prepared for a period of dormancy.

Vetivert, thank you so much for your reply! You Kiwi's are the best! I really don't need more begonias and would not feel badly if this one died but I tend to nurture all living things. I just want to play with the seed, I do so enjoy watching a seed awaken and grow.