It's been a while since I played harmonica but a C won't work for sure. You'll need either a E harmonica to play first position (or Straight harp), or an A harmo to play second position (blues style). Second position (cross harp), is a fourth higher than your tonality (that's five semitones).
a key chart for harmonica
The first position playing sound major and is probably what you would want to do to sound like Bob Dylan. Cross harp involves mainly inhaling and a lot of bending and would sound a
lot more bluesy (it's also a lot harder).
EDIT : I've given more thought to it and as Dave Wilson stated it I think I messed up the thinking.
It's C#m so E(maj), so either an harmo in E to play first position or an harmo in A to play cross harp.
I'm no Bob Dylan/harmonica specialist but it should guide you in the right direction.