In　this　paper,　a　new　extended　(3　+　1)-dimensional　generalized　Kadomtsev-Petviashvili　(KP)-Boussinesq　equation,　with　two　additional　terms　utt$$　{u}_{tt}　$$　and　uxz$$　{u}_{xz}　$$,　is　proposed　and　investigated.　This　new　equation　models　both　left　and　right　going　waves　like　the　Boussinesq　equation　and　describes　the　transmission　of　tsunami　waves　at　the　bottom　of　the　ocean　and　nonlinear　ion-acoustic　waves　in　the　magnetized　dusty　plasm.　We　have　constructed　one　set　of　breathers　and　first　order　periodic　waves　from　the　two-soliton　solution.　Moreover,　the　four-soliton　solution　consists　of　two　sets　of　breathers,　second　order　periodic　waves,　and　a　hybrid　of　breathers　and　periodic　line　waves.　Then,　we　introduce　two　kinds　of　＂long　wave＂　limits　to　obtain　the　rational　and　semi-rational　solutions　for　N=2,3,4$$　N=2,3,4　$$.　Under　specific　parametric　constraints,　the　obtained　rational　and　semi-rational　solutions　are　nonsingular.　The　rational　solution　can　be　classified　as　first　order　line　rogue　wave,　single　breather,　second　order　line　rogue　wave,　double　breather,　a　hybrid　of　breather　and　line　rogue　wave,　a　hybrid　of　breather,　and　single　soliton.　The　semi-rational　solution　can　be　classified　as　the　first　and　second　order　kink-shaped　rogue　wave,　a　hybrid　of　breather　and　one　(two)　soliton(s),　a　hybrid　of　a　set　of　breathers,　and　a　single　soliton　(breather).　In　addition,　we　give　Theorem　2.1　for　the　higher　order　rational　solutions.