  The Teichmueller space Teich(S) of a surface S in genus g>1 is a totally real
submanifold of the quasifuchsian space QF(S). We show that the determinant of
the Laplacian det'(\Delta) on Teich(S) has a unique holomorphic extension to
QF(S). To realize this holomorphic extension as the determinant of differential
operators on S, we introduce a holomorphic family {\Delta_{\mu,\nu}} of
elliptic second order differential operators on S whose parameter space is the
space of pairs of Beltrami differentials on S and which naturally extends the
Laplace operators of hyperbolic metrics on S. We study the determinant of this
family {\Delta_{\mu,\nu}} and show how this family realizes the holomorphic
extension of det'(\Delta) as its determinant.
