This tower is one end of the old Church of Mary Magdalene. The rest of the church (save foundations) was destroyed. These days the structure goes by the name “Buda Tower” because it’s a tourist attraction one can use to see the city from on high. The structure dates back to the 13th century.
They’re building a new park area at the base of Buda Castle, in front of the Palace. It was open while we were in town for the holidays. (Apparently, it has opened and closed several times and is still not entirely complete–though it does look nice.) The opened-closed thing being tied to the electoral cycle. Though apparently a significant portion of the funds came from the EU rather than domestic tax revenues.