  We introduce mutually unbiased complex Hadamard (MUCH) matrices and show that
the number of MUCH matrices of order 2n, n odd, is at most 2 and the bound is
attained for n = 1,5,9. Furthermore, we prove that certain pairs of mutually
unbiased complex Hadamard matrices of order m can be used to construct pairs of
unbiased real Hadamard matrices of order 2m. As a consequence we generate a new
pair of unbiased real Hadamard matrices of order 36.
