This paper addresses the problem of joint tracking and classification (JTC) of a single extended target with a complex shape. To describe this complex shape, the spatial extent state is first modeled by star-convex shape via a random hypersurface model (RHM), and then used as feature information for target classification. The target state is modeled by two vectors to alleviate the influence of the high-dimensional state space and the severely nonlinear observation model on target state estimation, while the Euclidean distance metric of the normalized Fourier descriptors is applied to obtain the analytical solution of the updated class probability. Consequently, the resulting method is called the “JTC-RHM method.” Besides, the proposed JTC-RHM is integrated into a Bernoulli filter framework to solve the JTC of a single extended target in the presence of detection uncertainty and clutter, resulting in a JTC-RHM-Ber filter. Specifically, the recursive expressions of this filter are derived. Simulations indicate that: (1) the proposed JTC-RHM method can classify the targets with complex shapes and similar sizes more correctly, compared with the JTC method based on the random matrix model; (2) the proposed method performs better in target state estimation than the star-convex RHM based extended target tracking method; (3) the proposed JTC-RHM-Ber filter has a promising performance in state detection and estimation, and can achieve target classification correctly.