  As the third stage of the project multi-indexed orthogonal polynomials, we
present, in the framework of 'discrete quantum mechanics' with pure imaginary
shifts in one dimension, the multi-indexed Wilson and Askey-Wilson polynomials.
They are obtained from the original Wilson and Askey-Wilson polynomials by
multiple application of the discrete analogue of the Darboux transformations or
the Crum-Krein-Adler deletion of 'virtual state solutions' of type I and II, in
a similar way to the multi-indexed Laguerre, Jacobi and (q-)Racah polynomials
reported earlier.
