  We study Motzkin paths of length $L$ with general weights on the edges and
end points. We investigate the limit behavior of the initial and final segments
of the random Motzkin path viewed as a pair of processes starting from each of
the two end points as $L$ becomes large. We then study macroscopic limits of
the resulting processes, where in two different regimes we obtain Markov
processes that appeared in the description of the stationary measure for the
KPZ equation on the half line and of conjectural stationary measure of the
hypothetical KPZ fixed point on the half line. Our results rely on the behavior
of the Al-Salam--Chihara polynomials in the neighbourhood of the upper end of
their orthogonality interval and on the limiting properties of the
$q$-Pochhammer and $q$-Gamma functions as $q\nearrow 1$.
