In this paper, we investigate from a theoretical point of view the 2D reaction-diffusion system for electrodeposition coupling morphology and surface chemistry, presented and experimentally validated in Bozzini et al. (2013J. Solid State Electr.17, 467–479). We analyse the mechanisms responsible for spatio-temporal organization. As a first step, spatially uniform dynamics is discussed and the occurrence of a supercritical Hopf bifurcation for the local kinetics is proved. In the spatial case, initiation of morphological patterns induced by diffusion is shown to occur in a suitable region of the parameter space. The intriguing interplay between Hopf and Turing instability is also considered, by investigating the spatio-temporal behaviour of the system in the neighbourhood of the codimension-two Turing--Hopf bifurcation point. An ADI (Alternating Direction Implicit) scheme based on high-order finite differences in space is applied to obtain numerical approximations of Turing patterns at the steady state and for the simulation of the oscillating Turing–Hopf dynamics.