  We construct ladder operators, $\tilde{C}$ and $\tilde{C^\dagger}$, for a
multi-step rational extension of the harmonic oscillator on the half plane,
$x\ge0$. These ladder operators connect all states of the spectrum in only
infinite-dimensional representations of their polynomial Heisenberg algebra.
For comparison, we also construct two different classes of ladder operator
acting on this system that form finite-dimensional as well as
infinite-dimensional representations of their respective polynomial Heisenberg
algebras. For the rational extension, we construct the position wavefunctions
in terms of exceptional orthogonal polynomials. For a particular choice of
parameters, we construct the coherent states, eigenvectors of $\tilde{C}$ with
generally complex eigenvalues, $z$, as superpositions of a subset of the energy
eigenvectors. Then we calculate the properties of these coherent states,
looking for classical or non-classical behaviour. We calculate the energy
expectation as a function of $|z|$. We plot position probability densities for
the coherent states and for the even and odd cat states formed from these
coherent states. We plot the Wigner function for a particular choice of $z$.
For these coherent states on one arm of a beamsplitter, we calculate the two
excitation number distribution and the linear entropy of the output state. We
plot the standard deviations in $x$ and $p$ and find no squeezing in the regime
considered. By plotting the Mandel $Q$ parameter for the coherent states as a
function of $|z|$, we find that the number statistics is sub-Poissonian.
