As the existing complex number Nth root computation methods are relatively discrete, we propose a general method and architecture based on coordinate rotation digital computer (CORDIC) to compute arbitrary complex number Nth root for the first time. Our method performs the tasks of complex modulus, complex phase angle, real Nth root, sine function and cosine function, which can be implemented by circular CORDIC , linear CORDIC and hyperbolic CORDIC. Based on these CORDICs, our proposed architecture can not only improve the hardware efficiency just through shift-add operations, but also flexibly adjust the precision and the input range of complex number Nth root. To prove its feasibility, we conduct a software simulation and implement an example circuit in hardware. Under the TSMC 28nm CMOS technology, we synthesize it and get the report that it has the area of 6561um^2 and the power of 3.95mW at the frequency of 1.5GHz.