Fractional calculus in complex analysis According to Fractional calculus, we know that $$(J^\alpha f) ( x ) = { 1 \over \Gamma ( \alpha ) } \int_0^x (x-t)^{\alpha-1} f(t) \; dt$$ 

It's in real analysis, but what about in complex analysis? As we know, if $\alpha$ is not a integer, then $(x-t)^{\alpha-1}$ may returns more than one values. 

So my question is can we find a method to let fractional calculus working on complex field?
There are a lot of definitions of fractional derivatives. If you want an (almost complete) answer for your question, then try the book Samko, Kilbas, Marichev, Fractional integrals and derivatives: theory and applications (1993). Specially, Ch 4, § 22 Fractional Integrals and Derivatives 
in the Complex Plane . "We emphasize that any work with definitions  requires precision aimed to single out a branch of the multivalued function. It is usually achieved by means of a cut which goes from the branching point to infinity or by fixing $\arg(t - z)$ in one or another way. Different choices of a cut, which fixes the branch 
of the function $(t - z)^{1+\alpha}$ , and of the curve, gives different values of 
$f^{(\alpha)}(z)$ in general.
Its easy. Just do the integral and then just replace your real x with complex z.