Quaternion　algebra　H　is　a　noncommutative　associative　algebra.　In　recent　years,　quaternionic　Fourier　analysis　has　received　increasing　attention　due　to　its　applications　in　signal　analysis　and　image　processing.　This　paper　addresses　conjugate　phase　retrieval　problem　in　the　quaternion　Euclidean　space　H-M　with　M　＞=　2.　Write　C-eta　=　{xi　:　xi　=　xi(0)　+　beta　eta,　xi(0),　beta　is　an　element　of　R}　for　eta　is　an　element　of　{i,　j,　k}.　We　remark　that　not　only　C-eta(M)-vectors　cannot　allow　traditional　conjugate　phase　retrieval　in　H-M,　but　also　C-i(i)M　boolean　OR　C-j(M)-complex　vectors　cannot　allow　phase　retrieval　in　H-M　.　We　are　devoted　to　conjugate　phase　retrieval　of　C-i(M)　boolean　OR　C-j(M)　-complex　vectors　in　H-M,　where　＂conjugate＂　is　not　the　traditional　conjugate.　We　introduce　the　notions　of　conjugation,　maximal　commutative　subset　and　conjugate　phase　retrieval.　Using　the　phase　lifting　techniques,　we　present　some　sufficient　conditions　on　complex　vectors　allowing　conjugate　phase　retrieval.　And　some　examples　are　also　provided　to　illustrate　the　generality　of　our　theory.