Mutually unbiased unit Hadamard (MUUH) matrices have been studied for almost 40 years. Recent interest in such matrices has been motivated by their applications to quantum information theory. In this paper, we introduce the class of mutually unbiased complex Hadamard (MUCH) matrices having as sntries fourth roots of unity. The number of MUCH matrices of order 2n, n odd, is at most 2, and the bound is attained for n = 1,5,9. 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. We then turn our attention to the class of mutually unbiased real Hadamard matrices of order 16n² which seem to be the most interesting ones. Mutually unbiased weighing (MUW) matrices are introduced for the first time. Further study of these objects look quite promising.