The breakup of compound jets into compound drops is of immense practical importance as the phenomenon can be exploited to make thin-walled spherical shells or hollow spheres that can be used for the controlled release of drugs and as fusion targets, among others. Finite amplitude deformation and breakup of a compound jet, consisting of a core and a shell that are both incompressible Newtonian fluids and each free surface of which is covered with a monolayer of an insoluble surfactant, that is surrounded by a passive gas is analyzed computationally by a temporal analysis. The 3D axisymmetric or 2D Navier Stokes system of equations together with the equations governing surfactant transport are solved by a method of lines algorithm in which the Galerkin/finite element method with elliptic mesh generation is used for spatial discretization and an adaptive finite difference method is employed for time integration. The dynamics are initiated by subjecting the jet to one of two possible sets of initial perturbations. In the first, the inner and the outer interfaces of a quiescent compound jet that is uniformly covered with a fixed amount of surfactant is subjected to axially periodic shape perturbations that are either in phase, w=0, or 180 degrees out of phase, w=180 degrees, where w is the phase shift between the disturbances imposed on the two interfaces. In the second, the initial distribution of surfactant along the two cylindrical or slightly perturbed interfaces are taken to vary in an axially periodic fashion such that the surfactant concentration perturbations are either in phase, s=0, or 180 degrees out phase, s=180 degrees, where s is the phase shift between the surfactant and shape perturbations on the liquid-liquid or the liquid-gas interface. Computations reveal that recirculating flows occur commonly during the deformation and pinch-off of compound jets, and hence render inapplicable the use of slender-jet type approximations for analyzing the dynamics in such cases. Moreover, as the deformations of one or both of the interfaces of the compound jet grow, the resulting shapes at the incipience of pinch-off are asymmetric and lead to the formation of satellite drops. Calculations are carried out over a wide range of Reynolds numbers of the core fluid, ratios of the viscosity and density of the shell fluid to those of the core fluid, ratio of the surface tension of the outer interface to the interfacial tension of the inner interface, the ratio of the unperturbed radius of the outer cylindrical interface to that of the inner cylindrical interface, wavenumber, Peclet number of the core fluid, ratios of surface diffusivity and maximum surface coverage of the shell fluid to those of the core fluid, initial values of surfactant coverage on the two interfaces, and amplitudes of the shape and surfactant perturbations to determine their effects on breakup time and whether both interfaces pinch at the same instant in time to result in the formation of compound drops.