This paper is concerned with a characterization of the observability for a continuous-time hidden Markov model where the state evolves as a general continuous-time Markov process and the observation process is modeled as nonlinear function of the state corrupted by the Gaussian measurement noise. The main technical tool is based on the recently discovered duality relationship between minimum variance estimation and stochastic optimal control: The observability is defined as a dual of the controllability for a certain backward stochastic differential equation. Based on the dual formulation, a test for observability is presented and related to literature. The proposed duality-based framework allows one to easily relate and compare the linear and the nonlinear systems. A side-by-side summary of this relationship is given in a tabular form (Table~1)