While error estimate for C1 finite element (FE) analysis is relatively rare in engineering domain, the present paper proposes a simple yet effective method based on the reduced element technique. In the present method, encouraged by the successful performance of the reduced element in general C0 FE analysis, a carefully chosen pair of C1 triangular elements, i.e., the Argyris triangle and the Bell triangle, are interwoven to form the so-called reduced element for point-wise error estimation, which paves the way for future possible adaptive analysis of thin plate bending problems in maximum norm. Specifically, the FE analysis is made as usual by using the Argyris element, and then after obtaining the FE results, all terms in the Bell element are extracted to form the reduced element solution and the remaining higher-order terms are served as the point-wise error estimator to estimate the errors in the reduced element. Numerous numerical experiments presented in this paper demonstrate the convergence orders of the nodal solution and the element interior solution in maximum norm for the Argyris, Bell, and reduced element respectively, verifying the feasibility and effectiveness of the proposed method. The prospective application of this reduced element to adaptive FE analysis is briefly discussed.