In light of recent interest in Hadamard diagonalisable graphs (graphs whose Laplacian matrix is diagonalisable by a Hadamard matrix), we generalise this notion from real to complex Hadamard matrices. We give some basic properties and methods of constructing such graphs. We show that a large class of complex Hadamard diagonalisable graphs have vertex sets forming an equitable partition, and that the Laplacian eigenvalues must be even integers. We provide a number of examples and constructions of complex Hadamard diagonalisable graphs, including two special classes of graphs: the Cayley graphs over $\mathbb{Z}_r^d$, and the non--complete extended $p$--sum (NEPS). We discuss necessary and sufficient conditions for $(α, β)$--Laplacian fractional revival and perfect state transfer on continuous--time quantum walks described by complex Hadamard diagonalisable graphs and provide examples of such quantum state transfer.