The proximal point algorithm (PPA) has been widely used in convex optimization. Many algorithms fall into the framework of PPA. To guarantee the convergence of the PPA, however, existing results conventionally need to ensure the positive definiteness of the corresponding proximal measure. In some senses, this essentially results in tiny step sizes (or over regularization) for the subproblems and thus inevitably decelerates the overall convergence speed of the PPA. In this paper, we investigate the possibility of relaxing the positive definiteness requirement of the proximal measure in PPA. An indefinite PPA for finding a zero of maximal monotone operator is thus proposed via choosing an indefinite proximal regularization term, resulting in larger step sizes. Under some suitable conditions, the global convergences of the proposed algorithm and its extension are proved. To make our method more practical, we suggested to solve the subproblem in an approximate manner and propose two flexible inexact criteria. We show the condition which guarantees the convergence of the proposed indefinite PPA is tight by a simple example. Finally, some preliminary numerical experiments are performed to show the efficiencies of the proposed algorithms.