In　this　paper,　we　accomplish　the　unified　convergence　analysis　of　a　second-order　method　of　multipliers　(i.e.,　a　second-order　augmented　Lagrangian　method)　for　solving　the　conventional　nonlinear　conic　optimization　problems.　Specifically,　the　algorithm　that　we　investigate　incorporates　a　specially　designed　nonsmooth　(generalized)　Newton　step　to　furnish　a　second-order　update　rule　for　the　multipliers.　We　first　show　in　a　unified　fashion　that　under　a　few　abstract　assumptions,　the　proposed　method　is　locally　convergent　and　possesses　a　(nonasymptotic)　superlinear　convergence　rate,　even　though　the　penalty　parameter　is　fixed　and/or　the　strict　complementarity　fails.　Subsequently,　we　demonstrate　that　for　the　three　typical　scenarios,　i.e.,　the　classic　nonlinear　programming,　the　nonlinear　second-order　cone　programming　and　the　nonlinear　semidefinite　programming,　these　abstract　assumptions　are　nothing　but　exactly　the　implications　of　the　iconic　sufficient　conditions　that　are　assumed　for　establishing　the　Q-linear　convergence　rates　of　the　method　of　multipliers　without　assuming　the　strict　complementarity.