Find Series[x^(1/2)D] then multiply by x^(-1/2):
In[1]:= Series[
x^(1/2) ParabolicCylinderD[-(1/2), (-1 + I) x], {x, \[Infinity], 0}]
In[2]:= Series[
x^(1/2) ParabolicCylinderD[-(1/2), (-1 + I) x], {x, \[Infinity], 1}]
You get same result for both inputs.
