  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 $(\alpha, \beta)$--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.
