We give the first natural examples of Calder\'on-Zygmund operators in the
theory of analysis on post-critically finite self-similar fractals. This is
achieved by showing that the purely imaginary Riesz and Bessel potentials on
nested fractals with 3 or more boundary points are of this type. It follows
that these operators are bounded on $L^{p}$, $1<p<\infty$ and satisfy weak 1-1
bounds. The analysis may be extended to infinite blow-ups of these fractals,
and to product spaces based on the fractal or its blow-up.