In this article, the use of channel state information (CSI) for indoor positioning is studied. In the considered model, a server equipped with several antennas sends pilot signals to users, while each user uses the received pilot signals to estimate channel states for user positioning. To this end, we formulate the positioning problem as an optimization problem aiming to minimize the gap between the estimated positions and the ground truth positions of users. To solve this problem, we design a complex-valued neural network (CVNN) model based federated learning (FL) algorithm. Compared to standard real-valued centralized machine learning (ML) methods, our proposed algorithm has two main advantages. First, our proposed algorithm can directly process complex-valued CSI data without data transformation. Second, our proposed algorithm is a distributed ML method that does not require users to send their CSI data to the server. Since the output of our proposed algorithm is complex-valued which consists of the real and imaginary parts, we study the use of the CVNN to implement two learning tasks. First, the proposed algorithm directly outputs the estimated positions of a user. Here, the real and imaginary parts of an output neuron represent the 2D coordinates of the user. Second, the proposed method can output two CSI features (i.e., line-of-sight/non-line-of-sight transmission link classification and time of arrival (TOA) prediction) which can be used in traditional positioning algorithms. Simulation results demonstrate that our designed CVNN based FL can reduce the mean positioning error between the estimated position and the actual position by up to 36%, compared to a RVNN based FL which requires to transform CSI data into real-valued data.