This work introduces an unconventional inexact augmented Lagrangian method where the augmenting term is a Euclidean norm raised to a power between one and two. The proposed algorithm is applicable to a broad class of constrained nonconvex minimization problems that involve nonlinear equality constraints. In a first part of this work, we conduct a full complexity analysis of the method under a mild regularity condition, leveraging an accelerated first-order algorithm for solving the Hölder-smooth subproblems. Interestingly, this worst-case result indicates that using lower powers for the augmenting term leads to faster constraint satisfaction, albeit with a slower decrease of the dual residual. Notably, our analysis does not assume boundedness of the iterates. Thereafter, we present an inexact proximal point method for solving the weakly-convex and Hölder-smooth subproblems, and demonstrate that the combined scheme attains an improved rate that reduces to the best-known convergence rate whenever the augmenting term is a classical squared Euclidean norm. Different augmenting terms, involving a lower power, further improve the primal complexity at the cost of the dual complexity. Finally, numerical experiments validate the practical performance of unconventional augmenting terms.