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It is shown that the basic Z_2-graded Lie superalgebras of the orthosymplectic and unitary series can be extended to Z_2x...xZ_2-graded superalgebras with the same even part of the Z_2-case. For this purpose, a Z_2x...xZ_2-graded bosonic algebra is introduced using a cliffonic dressing of an ordinary bosonic algebra. Some real forms of such Z_2x...xZ_2-superalgebras are considered. A new concept of Hermitian imaginary unit is introduced (the usual imaginary unit is an anti-Hermitian one).