  We study the continuous time quantum walk of a single particle (initially
localized at a single site) on a one-dimensional spatial lattice with complex
nearest neighbour and next-nearest neighbour hopping amplitudes. Complex
couplings lead to chiral propagation and a causal cone structure asymmetric
about the origin. We provide a hydrodynamic description for quantum walk
dynamics in large space time limit. We find a global "quasi-stationary state"
which can be described in terms of the local quasi-particle densities
satisfying Euler type of hydrodynamic equation and is characterized by an
infinite set of conservation laws satisfied by scaled cumulative position
moments. Further, we show that there is anomalous sub-diffusive scaling near
the extremal fronts, which can be described by higher order hydrodynamic
equations. The long time behaviour for any complex next-nearest neighbour
hopping with a non-zero real component is similar to that of purely real
hopping (apart from asymmetric distribution). There is a critical coupling
strength at which there is a Lifshitz transition where the topology of the
causal structure changes from a regime with one causal cone to a regime with
two nested causal cones. On the other hand, for purely imaginary next-nearest
neighbour hopping, there is a transition from one causal cone to a regime with
two partially overlapping cones due to the existence of degenerate maximal
fronts (moving with the same maximal velocity). The nature of the Lifshitz
transition and the scaling behaviour (both) at the critical coupling strength
is different in the two cases.
