  We give a construction of quasiminimal fields equipped with pseudo-analytic
maps, generalising Zilber's pseudo-exponential function. In particular we
construct pseudo-exponential maps of simple abelian varieties, including
pseudo-$\wp$-functions for elliptic curves. We show that the complex field with
the corresponding analytic function is isomorphic to the pseudo-analytic
version if and only the appropriate version of Schanuel's conjecture is true
and the corresponding version of the strong exponential-algebraic closedness
property holds.
  Moreover, we relativize the construction to build a model over a fairly
arbitrary countable subfield and deduce that the complex exponential field is
quasiminimal if it is exponentially-algebraically closed. This property asks
only that the graph of exponentiation have non-trivial intersection with
certain algebraic varieties but does not require genericity of these points.
Furthermore Schanuel's conjecture is not required as a condition for
quasiminimality.
