  CoNb$_2$O$_6$ is one of the few materials that is known to approximate the
one-dimensional transverse-field Ising model (1D-TFIM) near its quantum
critical point. It has been inferred that Co$^{2+}$ acts as a pseudo-spin 1/2
with anisotropic exchange interactions that are largely Ising-like, enabling
the realization of the TFIM. However, the behavior of CoNb$_2$O$_6$ is known to
diverge from the ideal TFIM under transverse magnetic fields that are far from
the quantum critical point, requiring the consideration of additional
anisotropic, bond-dependent (Kitaev-like) terms in the microscopic pseudo-spin
1/2 Hamiltonian. These terms are expected to be controlled in part by
single-ion physics, namely the wavefunction for the pseudo-spin 1/2 angular
momentum doublet. Here, we present the results of both inelastic neutron
scattering measurements and electron paramagnetic resonance spectroscopy on
CoNb$_2$O$_6$, which elucidate the single-ion physics of Co$^{2+}$ in
CoNb$_2$O$_6$ for the first time. We find that the system is well-described by
an intermediate spin-orbit coupled Hamiltonian, and the ground state is a
well-isolated Kramers doublet with an anisotropic $g$-tensor. We provide the
approximate wavefunctions for this doublet, which we expect will be useful in
theoretical investigations of the anisotropic exchange interactions.
