Bush-type Butson Hadamard matrices are introduced. It is shown that a nonextendable set of mutually unbiased Butson Hadamard matrices is obtained by adding a specific Butson Hadamard matrix to a set of mutually unbiased Bush-type Butson Hadamard matrices. A class of symmetric Bush-type Butson Hadamard matrices over the group G of n-th roots of unity is introduced that is also valid over any subgroup of G. The case of Bush-type Butson Hadamard matrices of even order will be discussed.