This paper develops a framework for the Hamiltonian quantization of complex Chern-Simons theory with gauge group \(\text{SL}(2,{\mathbb{C}})\) at an even level \(k\in {\mathbb{Z}}_{+}\) . Our approach follows the procedure of combinatorial quantization to construct the operator algebras of quantum holonomies on 2-surfaces and develop the representation theory. The *-representation of the operator algebra is carried by the infinite dimensional Hilbert space \({\mathcal{H}}_{\overrightarrow{\lambda }}\) and closely connects to the infinite-dimensional *-representation of the quantum deformed Lorentz group \({\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)\) . The quantum group \({\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)\) also emerges from the quantum gauge transformations of the complex Chern-Simons theory. Focusing on a m-holed sphere Σ0,m, the physical Hilbert space \({\mathcal{H}}_{\text{phys}}\) is identified by imposing the gauge invariance and the flatness constraint. The states in \({\mathcal{H}}_{\text{phys}}\) are the \({\mathcal{U}}_{\text{q}}\left(s{l}_{2}\right)\otimes {\mathcal{U}}_{\widetilde{\text{q}}}\left(s{l}_{2}\right)\) -invariant linear functionals on a dense domain in \({\mathcal{H}}_{\overrightarrow{\lambda }}\) . Finally, we demonstrate that the physical Hilbert space carries a Fenchel-Nielsen representation, where a set of Wilson loop operators associated with a pants decomposition of Σ0,m are diagonalized.
