Recently, the transition metal dichalcogenides have attracted renewed attention due to the potential use of their low-dimensional forms in both nano- and opto-electronics. In such applications, the electronic and transport properties of monolayer transition metal dichalcogenides play a pivotal role. The present paper provides a new insight into these essential properties by studying the complex band structures of popular transition metal dichalcogenide monolayers (MX 2, where M = Mo, W; X = S, Se, Te) while including spin-orbit coupling effects. The conducted symmetry-based tight-binding calculations show that the analytical continuation from the real band structures to the complex momentum space leads to nonlinear generalized eigenvalue problems. Herein an efficient method for solving such a class of nonlinear problems is presented and yields a complete set of physically relevant eigenvalues. Solutions obtained by this method are characterized and classified into propagating and evanescent states, where the latter states manifest not only monotonic but also oscillatory decay character. It is observed that some of the oscillatory evanescent states create characteristic complex loops at the direct band gap of MX 2 monolayers, where electrons can directly tunnel between the band gap edges. To describe these tunneling currents, decay behavior of electronic states in the forbidden energy region is elucidated and their importance within the ballistic transport regime is briefly discussed.