Is the DFT matrix the unique* unitary matrix with all entries of same magnitude?
(*up to some trivial transformations)
Call a unitary matrix flat if all its entries have the same absolute value. In operator theory these arise as a class of type-II matrices, which were used by Vaughan Jones in his work on link invariants. Currently they are also of interest in physics, because of their connection with "mutually unbiased bases". In this context they are known as generalized Hadamard matrices (which is a good enough name, but has a different meaning among design theorists).
The basic examples are Hadamard matrices and character tables of abelian groups, as noted by Mark and Steve. The class of flat unitary matrices is closed under Kronecker product, and this gives us examples which are neither Hadamard nor character tables. A survey of the subject by physicists appears as arXiv:quant-ph/0512154. (And if you search on quant-ph for articles with "Hadamard" in the title, you'll find many more papers on the subject.)