  We extend the complex-valued analytic torsion, introduced by Burghelea and
Haller on closed manifolds, to compact Riemannian bordisms. We do so by
considering a flat complex vector bundle over a compact Riemannian manifold,
endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian
metric and the bilinear form are used to define non-selfadjoint Laplacians
acting on vector-valued smooth forms under absolute and relative boundary
conditions. In the process to define the complex-valued analytic torsion, we
study spectral properties associated to these generalized Laplacians. As main
results, we obtain anomaly formulas for the complex-valued analytic torsion.
Our reasoning takes into account that the coefficients in the heat trace
asymptotic expansion associated to the boundary value problem under
consideration, are locally computable. The anomaly formulas for the
complex-valued Ray--Singer torsion are obtained by using the corresponding ones
for the Ray--Singer metric, obtained by Bruening and Ma on manifolds with
boundary, and an argument of analytic continuation. In odd dimensions, our
anomaly formulas are in accord with the corresponding results of Su, without
requiring the variations of the Riemannian metric and bilinear structures to be
supported in the interior of the manifold.

Differential Geometry