We consider the determinant of the Laplacian as a function on the Teichmueller space of a closed surface S of genus > 1 and show that it has a unique holomorphic extension to the quasifuchsian space. To realize this holomorphic extension as determinants of actual differential operators on S, we construct holomorphic extensions of Laplace operators of hyperbolic metrics on S using Ahlfors-Bers theory of quasiconformal mappings, and we study their determinants.