  We explore the dynamical behavior at and near a special class of
two-dimensional quantum critical points. Each is a conformal quantum critical
point (CQCP), where in the scaling limit the equal-time correlators are those
of a two-dimensional conformal field theory. The critical theories include the
square-lattice quantum dimer model, the quantum Lifshitz theory, and a deformed
toric code model. We show that under generic perturbation the latter flows
toward the ordinary Lorentz-invariant (2+1) dimensional Ising critical point,
illustrating that CQCPs are generically unstable. We exploit a correspondence
between the classical and quantum dynamical behavior in such systems to perform
an extensive numerical study of two lines of CQCPs in a quantum eight-vertex
model, or equivalently, two coupled deformed toric codes. We find that the
dynamical critical exponent z remains 2 along the U(1)-symmetric quantum
Lifshitz line, while it continuously varies along the line with only Z_2
symmetry. This illustrates how two CQCPs can have very different dynamical
properties, despite identical equal-time ground-state correlators. Our results
equally apply to the dynamics of the corresponding purely classical models.
