The outdoor pools at Széchenyi stay open all winter, and on a cold December morning in City Park they nearly disappear under their own steam. That sight is the strongest case for Budapest in December. The days are short and the wind off the Danube has an edge, but the city sits on hot springs, and a slow trip suits the month. We would give each day one bath and one market, and walk everywhere in between.
Széchenyi is the one everyone photographs. Rudas is the better bath for a long afternoon. It sits at the foot of Gellért Hill on the Buda bank, and its heart is sixteenth-century Ottoman work, an octagonal pool under a dome pierced with small windows, where daylight comes down through the steam in shafts. The Turkish section is single-sex for part of most weekdays and mixed at weekends, so check the schedule before you cross the river.
Keep the markets for after dark. The largest fills Vörösmarty tér, in front of the Gerbeaud café where Váci utca ends. The prettier one wraps around St Stephen's Basilica, with a skating rink circling the tree and light projections moving over the façade in the evening. Both run right through December. Walk between the two, through the streets of Lipótváros. It takes under ten minutes, and it is the part of the evening we would least want to hurry.
On Friday and Saturday nights Rudas opens again late and stays open into the small hours. That is the time for its rooftop pool: hot water on a cold roof, with the river below and the lit bridges running across it to Pest.