  In this paper we investigate the global behavior of proper holomorphic maps f
from the unit disc U={|z|<1} to C^2. The fact that U is transcendental imposes
certain restrictions on the image f(U). For instance, f(U) cannot be contained
in any proper complex cone in C^2 since this would force it to be algebraic. On
the other hand, we show that a real cone in C^2 with axis R^2 contains the
image of a proper holomorphic map f from U to C^2 if and only if the angle of
the cone is larger than pi/2. We also construct maps f as above whose images
avoid both coordinate axes in C^2. Equivalently, we construct a pair of
positive harmonic functions u, v on U such that max{u(z),v(z)} tends to plus
infinity when z tends to the boundary of U. Furthermore we show that the
components f_1, f_1 of a proper holomorphic map from U to C^2, as well as
polynomial and certain rational functions of f_1 and f_2, have the property
that their essential range at any boundary point of U omits at most a polar set
in C.
