  We consider the basilica Julia set of the polynomial $P(z)=z^{2}-1$ and
construct all possible resistance (Dirichlet) forms, and the corresponding
Laplacians, for which the topology in the effective resistance metric coincides
with the usual topology. Then we concentrate on two particular cases. One is a
self-similar harmonic structure, for which the energy renormalization factor is
2, the spectral dimension is $\log9/\log6$, and we can compute all the
eigenvalues and eigenfunctions by a spectral decimation method. The other is
graph-directed self-similar under the map $z\mapsto P(z)$; it has energy
renormalization factor $\sqrt2$ and spectral dimension 4/3, but the exact
computation of the spectrum is difficult. The latter Dirichlet form and
Laplacian are in a sense conformally invariant on the basilica Julia set.
