In this paper, we consider a complex-valued d-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion. This process has been introduced by the second author of the article, and both its real and imaginary parts, restricted on the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on the analytic fBm can be constructed for any value of the Hurst parameter in (0,1/2). This allows in particular to solve differential equations driven by this process in a neighborhood of 0 of the complex upper half-plane, thanks to a variant of the usual rough path theory due to Gubinelli.