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 Z^d_r , and the non–complete extended p–sum (NEPS). We discuss necessary and sufficient conditions for (\alpha, \beta)–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.