  S^4 is not a complex manifold, but it is sufficient to remove one point to
make it complex. Using supersymmetry methods, we show that the Dolbeault
complex (involving the holomorphic exterior derivative and its Hermitian
conjugate) can be perfectly well defined in this case. We calculate the
spectrum of the Dolbeault Laplacian. It involves 3 bosonic zero modes such that
the Dolbeault index on S^4\{.} is equal to 3.
