The rheological behaviour of dense suspensions of ideally conductive particles in the presence of both electric field and shear flow is studied using large-scale numerical simulations. Under the action of an electric field, these particles are known to undergo dipolophoresis (DIP), which is the combination of two nonlinear electrokinetic phenomena: induced-charge electrophoresis (ICEP) and dielectrophoresis (DEP). For ideally conductive particles, ICEP is predominant over DEP, resulting in transient pairing dynamics. The shear viscosity and first and second normal stress differences
$N_1$ and
$N_2$ of such suspensions are examined over a range of volume fractions
$15\,\% \leq \phi \leq 50\,\%$ as a function of Mason number
$Mn$, which measures the relative importance of viscous shear stress over electrokinetic-driven stress. For
$Mn < 1$ or low shear rates, the DIP is shown to dominate the dynamics, resulting in a relatively low-viscosity state. The positive
$N_1$ and negative
$N_2$ are observed at
$\phi < 30\,\%$, which is similar to Brownian suspensions, while their signs are reversed at
$\phi \ge 30\,\%$. For
$Mn \ge 1$, the shear thickening starts to arise at
$\phi \ge 30\,\%$, and an almost five-fold increase in viscosity occurs at
$\phi = 50\,\%$. Both
$N_1$ and
$N_2$ are negative for
$Mn \gg 1$ at all volume fractions considered. We illuminate the transition in rheological behaviours from DIP to shear dominance around
$Mn = 1$ in connection to suspension microstructure and dynamics. Lastly, our findings reveal the potential use of nonlinear electrokinetics as a means of active rheology control for such suspensions.