This paper proposes a new complex zeroing neural network (NCZNN) to solve a set of dynamic complex linear equations, which is an extension from the design idea of the real-valued zeroing neural network. Different from the previous complex ZNN (CZNN) model, which cannot process nonlinear activation functions and thus only converge in infinite time, a nonlinear sign-bi-power (SBP) activation function is explored to enable the proposed NCZNN model to converge within finite time in complex domain by using two different ways. One is to simultaneously activate the real part and the imaginary part of a complex number and the other is to activate the modulus of a complex number. In addition, the detailed theoretical analyses of the NCZNN model are provided according to these two processing ways, and the corresponding convergence upper bounds are analytically calculated. Two numerical experiments are conducted by using the NCZNN model and the CZNN model to solve a set of dynamic complex linear equations. Comparative results further prove that the NCZNN model has better convergence performance than the CZNN model. At last, the proposed method is applied to the motion tracking of a mobile manipulator, and simulative results verify the feasibility of our method in robotic applications.