We consider the problem of estimating the state of a system when measurement noise is a function of the system's state. We propose generalizations of the extended Kalman filter and the iterated extended Kalman filter that can be utilized when the state estimate distribution is approximately Gaussian. The state estimate is computed by an iterative root-searching method that maximizes a maximum likelihood function. The new filter allows for the consistent treatment of a class of control problem involving non-linear estimation from measurements with state-dependent noise. The effectiveness of the estimation algorithm is illustrated for a control problem with a mobile bearing-only sensor.