How to implement split-complex numbers?

Try this:

J /: Power[J, p_Integer?OddQ] := J
J /: Power[J, p_Integer?EvenQ] := 1


J^Range[-10, 10]

{1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1, J, 1}