  We show that for each fixed non-constant complex polynomial $P$ of the plane
there exists a homeomorphism $h$ such that $P\circ h$ is a Lipschitz quotient
mapping. This corrects errors in the construction given earlier by Johnson et.
al. [Michigan Math. J. $\textbf{47}$ (2000), 15-31]. Further we introduce a
stronger notion of pointwise co-Lipschitzness and characterise its equivalence
to the standard pointwise definition whilst also highlighting its relevance to
a long-standing conjecture concerning Lipschitz quotient mappings
$\mathbb{R}^n\to\mathbb{R}^n, n\geq 3$.
