  So far, Seshadri constants on abelian surfaces are completely understood only
in the cases of Picard number one and on principally polarized abelian surfaces
with real multiplication. Beyond that, there are partial results for products
of elliptic curves. In this paper, we show how to compute the Seshadri constant
of any nef line bundle on any abelian surface over the complex numbers. We
develop an effective algorithm depending only on the basis of the
N\'eron-Severi group to compute not only the Seshadri constants but also the
numerical data of their Seshadri curves. Access to the Seshadri curves allows
us to plot Seshadri functions and better understand their structure. We show
that already in the case of Picard number two the complexity of Seshadri
functions can vary to a great degree. Our results indicate that aside from
finitely many cases the complexity of the Seshadri function is at least as high
as in the Cantor function.
