  We show that a confluent case of the big q-Jacobi polynomials P_n(x;a,b,c;q),
which corresponds to a=b=-c, leads to a discrete orthogonality relation for
imaginary values of the parameter a (outside of its commonly known domain 0<a<
q^{-1}). Since P_n(x;q^\alpha, q^\alpha, -q^\alpha; q) tend to Gegenbauer (or
ultraspherical) polynomials in the limit as q->1, this family represents yet
another q-extension of these classical polynomials, different from the
continuous q-ultraspherical polynomials of Rogers. The dual family with respect
to the polynomials P_n(x;a,a,-a;q) (i.e., the dual discrete q-ultraspherical
polynomials) corresponds to the indeterminate moment problem, that is, these
polynomials have infinitely many orthogonality relations. We find orthogonality
relations for these polynomials, which have not been considered before. In
particular, extremal orthogonality measures for these polynomials are derived.
