Unless you plan some winter walking expeditions in the Alps, you don't need any special kind of boot or shoe, provided that it's at least close-toed. It may very well rain or snow during your visit, but it doesn't tend to come down very hard. More important is a pair of warm socks.
Let me share an example. In four years of living in Germany and hiking at least weekly during all seasons of the year, my shoes were soaked through on exactly two occassions. Once when I had to jump into a canal to rescue my dog and once when I got caught out in the open on top of a mountain during a sudden Alpine storm. Normal walking shoes were sufficient for all other occassions, with the exception of some Alpine winter walking, when I did wear snow boots.