  R.M. Kashaev conjectured that the asymptotic behavior of his link invariant,
which equals the colored Jones polynomial evaluated at a root of unity,
determines the hyperbolic volume of any hyperbolic link complement. We observe
numerically that for knots $6_3$, $8_9$ and $8_{20}$ and for the Whitehead
link, the colored Jones polynomials are related to the hyperbolic volumes and
the Chern-Simons invariants and propose a complexification of Kashaev's
conjecture.
