  Raised $k$-Dyck paths are a generalization of $k$-Dyck paths that may both
begin and end at a nonzero height. In this paper, we develop closed formulas
for the number of raised $k$-Dyck paths from $(0,\alpha)$ to $(\ell,\beta)$ for
all height pairs $\alpha,\beta \geq 0$, all lengths $\ell \geq 0$, and all $k
\geq 2$. We then enumerate raised $k$-Dyck paths with a fixed number of returns
to ground, a fixed minimum height, and a fixed maximum height, presenting
generating functions (in terms of the generating functions $C_k(t)$ for the
$k$-Catalan numbers) when closed formulas aren't tractable. Specializing our
results to $k=2$ or to $\alpha < k$ reveal connections with preexisting results
concerning height-bounded Dyck paths and "Dyck paths with a negative boundary",
respectively.
