  We consider noncompact open $SL(2, \mathbb{C})$ spin chain and construct
eigenfunctions of $B$-element of monodromy matrix for the simplest case of the
chain with one site. The reflection operator appearing in this construction can
be used to express eigenfunction for $n$ sites in terms of the eigenfunction
for $n-1$ sites, this general result is briefly announced. We prove
orthogonality and completeness of constructed eigenfunctions in the case of one
site, express them in terms of the hypergeometric function of the complex field
and derive the equation for the reflection operator with the general
$SL(2,\mathbb{C})$-invariant $\mathbb{R}$-operator.
