  We show that the index of a constant mean curvature 1 surface in hyperbolic
3-space is completely determined by the compact Riemann surface and secondary
Gauss map that represent it in Bryant's Weierstrass representation. We give
three applications of this observation. Firstly, it allows us to explicitly
compute the index of the catenoid cousins and some other examples. Secondly, it
allows us to be able to apply a method similar to that of Choe (using Killing
vector fields on minimal surfaces in Euclidean 3-space) to our case as well,
resulting in lower bounds of index for other examples. And thirdly, it allows
us to give a more direct proof of the result by do Carmo and Silveira that if a
constant mean curvature 1 surface in hyperbolic 3-space has finite total
curvature, then it has finite index. Finally, we show that for any constant
mean curvature 1 surface in hyperbolic 3-space that has been constructed via a
correspondence to a minimal surface in Euclidean 3-space, we can take advantage
of this correspondence to find a lower bound for its index.
