  We revisit the problem of determining the zero modes of the Dirac operator on
the Eguchi-Hanson space. It is well known that there are no normalisable zero
modes, but such zero modes do appear when the Dirac operator is twisted by a
$U(1)$ connection with $L^2$ normalisable curvature. The novelty of our
treatment is that we use the formalism of spin-$c$ spinors (or spinors as
differential forms), which makes the required calculations simpler. In
particular, to compute the Dirac operator we never need to compute the spin
connection. As a result, we are able to reproduce the known normalisable zero
modes of the twisted Eguchi-Hanson Dirac operator by relatively simple
computations. We also collect various different descriptions of the
Eguchi-Hanson space, including its construction as a hyperk\"ahler quotient of
$\mathbb{C}^4$ with the flat metric. The latter illustrates the geometric
origin of the connection with $L^2$ curvature used to twist the Dirac operator.
To illustrate the power of the formalism developed, we generalise the results
to the case of Dirac zero modes on the Ricci-flat K\"ahler manifolds obtained
by applying Calabi's construction to the canonical bundle of $\mathbb{C} P^n $.
