There exist many iterative algorithms for computing eigenvalues of large sparse matrices, e.g. Power, Lanczos, and Arnoldi iteration. Take the Lanczos algorithm for instance. It assumes that the matrix is Hermitian, in the standard inner product space. Since it only requires matrixvector multiplies and inner products as basic operations, is it possible to apply the algorithm to a nonHermitian matrix which is selfadjoint under a nonstandard inner product and still expect the algorithm to "work" in some sense? I'm imagining this in the case of discretizing selfadjoint operators with respect to a nonstandard bilinear form (not necessarily positive definite).

Using inner product $(x,My)$ in Lanczos is equivalent to using $M^{1}$ preconditioner in its preconditioned version. As far as I know, preconditioned version of Lanczos is used for generalized eigenproblems: $A x = \lambda M x$ Preconditioned Lanczos operates with two blackboxes: $Ax$ multiplier and $M^{1}x$ multiplier. 

