This is pretty neat! Here's why it works if anyone is interested:
Im[z] == (z - Conjugate[z])/(2I)
and thusÂ
Im[f[x + h*I]]/h == (f[x + h*I] - f[x - h*I])/(2h*I)
which is the central difference quotient.
(This only works if we assume x and h are real and f is analytic at x.)
