There are different possible definitions, the one ND uses is in terms of Cauchy integrals. For example, the 2nd derivative of z^4 at a=1 is
In[1]:= With[{n = 2, a = 1}, n!/(2 Pi) Integrate[z = a + Exp[I t]; z^4/(z - a)^n, {t, 0, 2 Pi}]]
Out[1]= 12
In this representation, let the derivative order n be I instead of 2:
In[2]:= With[{n = I, a = 1}, n!/(2 Pi) Integrate[z = a + Exp[I t]; z^4/(z - a)^n, {t, 0, 2 Pi}]]
Out[2]= -(12/85 - (48 I)/85) I! Sinh[Pi]/Pi
In[3]:= N[%]
Out[3]= 0.0632028 + 1.11425 I
