Filters relying on the Gaussian approximation typically incorporate the measurement linearly, i.e., the value of the measurement is pre-multiplied by a matrix-valued gain in the state update. Nonlinear filters that relax the Gaussian assumption, on the other hand, typically approximate the distribution of the state with a finite sum of point masses or Gaussian distributions. In this work, the distribution of the state is approximated by a polynomial transformation of a Gaussian distribution, allowing for all moments, central and raw, to be rapidly computed in closed form. Knowledge of the higher-order moments is then employed to perform a polynomial measurement update, i.e., the value of the measurement enters the update function as a polynomial of arbitrary order. A filter employing a Gaussian approximation with linear update is, therefore, a special case of the proposed algorithm when the order of the update is set to one. At the cost of more computations, the new methodology guarantees performance better than the linear/Gaussian approach for nonlinear systems. This work employs monomial basis functions and Taylor series, but it is readily extendable to an orthogonal polynomial basis.