We propose a method of complex-amplitude Fourier single-pixel imaging (CFSI) with coherent structured illumination to acquire both the amplitude and phase of an object. In the proposed method, an object is illustrated by a series of coherent structured light fields, which are generated by a phase-only spatial light modulator, the complex Fourier spectrum of the object can be acquired sequentially by a single-pixel photodetector. Then the desired complex-amplitude image can be retrieved directly by applying an inverse Fourier transform. We experimentally implemented this CFSI with several different types of objects. The experimental results show that the proposed method provides a promising complex-amplitude imaging approach with high quality and a stable configuration. Thus, it might find broad applications in optical metrology and biomedical science.