Heavy neutral Higgs boson production and decay into neutralino and chargino pairs is studied at the Large Hadron Collider in the context of the Minimal Supersymmetric Standard Model. Higgs boson decays into the heavier neutralino and chargino states, i.e., H^0 or A^0 to tilde{chi}_i^0 tilde{chi}_j^0 (i,j = 2,3,4) as well as H^0 or A^0 to tilde{chi}_1^{pm} tilde{chi}_2^{mp}, tilde{chi}_2^+ tilde{chi}_2^- (all leading to four-lepton plus missing transverse energy final states), is found to improve the possibilities of discovering such Higgs states beyond those previously identified by considering H^0 or A^0 to tilde{chi}_2^0 tilde{chi}_2^0 decays only. In particular, H^0,A^0 bosons with quite heavy masses, approaching ~800 GeV in the so-called `decoupling region where no clear SM signatures for the heavier MSSM Higgs bosons are known to exist, can now be discerned, for suitable but not particularly restrictive configurations of the low energy supersymmetric parameters. The high M_A discovery reach for the H^0 and A^0 may thus be greatly extended. Full event-generator level simulations, including realistic detector effects and analyses of all significant backgrounds, are performed to delineate the potential H^0,A^0 discovery regions. The wedgebox plot technique is also utilized to further analyze the 4l plus missing transverse energy signal and background events. This study marks the first thorough and reasonably complete analysis of this important class of MSSM Higgs boson signature modes. In fact, this is the first time discovery regions including all possible neutralino and chargino decay modes of the Higgs bosons have ever been mapped out.
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