  We present new exactly solvable systems of the discrete quantum mechanics
with pure imaginary shifts, whose physical range of the coordinate is the whole
real line. These systems are shape invariant and their eigenfunctions are
described by the multi-indexed continuous Hahn and Meixner-Pollaczek orthogonal
polynomials. The set of degrees of these multi-indexed polynomials are
$\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$,
where $\ell_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$ : a
multi-index set), but they form a complete set of orthogonal basis in the
weighted Hilbert space.
