  We study various aspects related to boundary regularity of complete properly
embedded Willmore surfaces in H3, particularly those related to assumptions on
boundedness or smallness of a certain weighted version of the Willmore energy.
We prove, in particular, that small energy controls C1 boundary regularity. We
examine the possible lack of C1 convergence for sequences of surfaces with
bounded Willmore energy and find that the mechanism responsible for this is a
bubbling phenomenon, where energy escapes to infinity.
