I'm not sure in which sense you come to the conclusion the Maxwell equations are incomplete or inconsistent. As far as phenomena are concerned that are describable in the classical approximation (i.e., neglecting quantum effects) the Maxwell equations are a complete description. The here discussed issues with "hidden momentum" are somewhat misleading, and I don't like this notion at all. There is no "hidden momentum". At the moment I'm writing on a little manuscript about this issue, where I try to avoid this idea of "hidden momentum". What's called "hidden momentum" is nothing else than a correct consideration of all sources of energies and momenta, including mechanical (kinetic), the electromagnetic, and the intrinsic stress of macroscopic bodies as an effective classical description of quantum phenomena (which on a microscopic level are also mostly electromagnetic in origin but go partially beyond the classical approximation, like the permanent magnetism caused by spins and many-body effects like exchange forces).
The apparent problems are from several sources of wrongly applied (mostly non-relativistic) approximations. One of the most simple cases is explained nicely in Griffiths's book on electrodynamics and illuminating in understanding the issue of so-called "hidden momentum" clearly. Unfortunately he expresses this issue in a somewhat oldfashioned way, because he uses the traditional way to present electromagnetism first in a kind of non-relativistic approximation as far as the mechanical part is concerned. This is almost always justified for everyday household currents but not always, and the example for the socalled "hidden momentum" he discusses seems to be pretty mysterious, but if you reformulate it only a bit using the exact relativistic expressions everywhere, all the mystery vanishes and it occurs that no part of the momentum was ever hiding somewhere, except in the sloppy mind of the physicist treating the charge carrier's mechanical momentum in the non-relativistic approximation, forgetting that the neglected terms are precisely the total momentum of the electromagnetic field which is precisely the opposite of the piece neglected in the non-relativistic approximation of the charge carriers' mechanical momentum. So let me reformulate the problem in the strictly relativistic form (although I'm using the non-covariant 3D treatment, which is more intuitive than the manifestly covariant 4D tensor formalism but nevertheless fully exact concerning relativistic effects).
He consideres a rectangular loop at rest carrying a steady current as a model for a magnetic dipole in an additional electrostatic field (which take as homogeneous across the loop) parallel to the vertical segments of the loop. The momenta of the charge carriers that make up the current in the two vertical pieces cancel but the momenta in the upper segment are different, because there is a change in energy due to the electrostatic potential \Phi=-E y. Let N_{<} be the (constant) number of charge carriers in the lower horizontal segment at y=0 and N_{>} the one at the upper segment at y=h. The total mechanical momentum of the charge carriers in the loop thus is (I set c=1 in this posting for simplicity)
\vec{p}_{\text{mech}}=(N_{>} E_{>} v_{>}-N_{<} E_{<} v_{<}) \vec{e}_x.
Now due to the stationary continuity equation \vec{\nabla} \cdot \vec{j}=0 the current is the same everywhere in the loop and thus
I=\frac{N_>}{l} Q v_>=\frac{N_<}{l} Q v_{<}
and thus
N_> v_>=N_< v_<=\frac{I l}{Q}.
Plugging this into the formula for the mechanical momentum
\vec{p}_{\text{mech}}=\frac{I l}{Q} (E_>-E_<) \vec{e}_x=I l E h \vec{e}_x,
because we have
E_>-E_<=(m+Q E h)-m=Q E h.
Now we need to evaluate the total field momentum. The momentum density is given by Poynting's vector, and thus the total momentum by
\vec{p}_{\text{em}}=\int \mathrm{d}^3 \vec{x} \vec{E} \times \vec{B}.
In our case, it's most easy to get this, if we could find an expression only involving the electric potential and the current density. We find such an expression by writing
(\vec{E} \times \vec{B})_j=-\epsilon_{jkl} (\partial_k \Phi) B_l = -\epsilon_{jkl} [\partial_k (\Phi B_l)-\Phi \partial_k B_l].
Integrating over the entire space gives 0 for the first term due to Stokes's integral theorem and the vanishing of the magnetic field at infinity and the second piece is
\vec{p}_{\text{em}}=\int \mathrm{d}^3 \vec{x} \Phi (\vec{\nabla} \times \vec{B}) = \int \mathrm{d}^3 \vec{x} \Phi \vec{j}.
The contributions from the vertical pieces of the loop cancel obviously. The constributions from the horizontal parts give
\vec{p}_{\text{em}}=-I E l h \vec{e}_x.
As we see, the total momentum is
\vec{p}_{\text{tot}}=\vec{p}_{\text{mech}}+\vec{p}_{\text{em}}=0,
as it must be due to the general theorem that in relativistic(!) physics any closed(!) system with a center of energy at rest must have total 0 momentum. Our closed system consists of the moving particles and the electromagnetic field and fufills the general theorem. An apparent paradox only occurs when one treats the momenta non-relativistically, which is wrong in this case no matter how slow the charge carriers might be when it comes to the balance between the mechanical and field momentum. The example also clearly shows that there is no mysterious "hidden momentum". It's only the wrong assumption we could use the non-relativistic approximation for the momentum of the charge carriers.
It's also very illuminating to think about the current as produced in an ideal-fluid picture. There it turns out that the "hidden momentum" occurs from the fact that the pressure has to be appropriately taken into account of the momentum in the upper and lower segment of the loop. Again, there's nothing mysterious or hidden about any part of the momentum, it's just the proper fully relativistic treatment of all parts of the setup.
There are a lot of similar examples. The historically most famous problem of this kind is the classical model for charged particles. To keep the charged particle stable one has to take into account the mechanical stresses holding the charges in place, because otherwise the like-sign charges would repel each other and the construct would simply blow appart (although even then there is no paradox if one can treat everything fully relativistically). The apparent paradox in this case was that the energy-momentum relation E^2-\vec{p}^2=m^2 for the model for a charged "particle" seemed to be violated, because one took the integral of the electromagnetic field energy and its momentum although for this tensor the equation of continuity doesn't hold and the fields alone do not form a closed system, but one has to take into account the charges and the mechanical stresses of their binding on a body to a static charge distribution. You find a very clear and very general treatment in Jackson, Classical Electrodynamics, 3rd edition, referring to a paper by Julian Schwinger, who wasn't only a master of quantum but also classical electrodynamics.
http://link.springer.com/article/10.1007/BF01906185
As in relativistic electrodynamics the mass of a particle is an empirical/phenomenological parameter which has to be adapted by tuning other parameters in the theory in the sense of "renormalization". Schwinger clearly shows that this is due to the ambiguity in defining the mechanical stresses needed to stabilize the particle (Poincare stresses) from considerations within electromagnetics alone.
