Conjugate is primarily for numbers and expressions that represent numbers (such as Root objects).
If you want Conjugate to work on the parameters inside an expression,
use Map with a level specification. For example,
In[1]:= Map[ Conjugate, Piecewise[{{-I, a > 0}}, I], {-1} ]
Out[1]= Piecewise[{{I, Conjugate[a] > 0}}, -I]
In[2]:= FullSimplify[ %, Element[a, Reals] ]
Out[2]= Piecewise[{{-I, a <= 0}}, I]