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High order finite difference WENO methods have the advantage of simpler coding and smaller computational cost for multi-dimensional problems, compared with finite volume WENO methods of the same order of accuracy. However a main restriction is that conservative finite difference methods of third and higher order of accuracy can only be used on uniform rectangular or smooth curvilinear meshes. In order to overcome this difficulty, in this paper we develop a multidomain high order WENO finite difference method which uses an interpolation procedure at the subdomain interfaces. A simple Lagrange interpolation procedure is implemented and compared to a WENO interpolation procedure. Extensive numerical examples are shown to indicate the effectiveness of each procedure, including the measurement of conservation errors, orders of accuracy, essentially non-oscillatory properties at the domain interfaces, and robustness for problems containing strong shocks and complex geometry. Our numerical experiments have shown that the simple and efficient Lagrange interpolation suffices for the subdomain interface treatment in the multidomain WENO finite difference method, to retain essential conservation, full high order of accuracy, essentially non-oscillatory properties at the domain interfaces even for strong shocks, and robustness for problems containing strong shocks and complex geometry. The method developed in this paper can be used to solve problems in relatively complex geometry at a much smaller CPU cost than the finite volume version of the same method for the same accuracy. The method can also be used for high order finite difference ENO schemes and an example is given to demonstrate a similar result as that for the WENO schemes.