I took a look at this a while back, and got as far as writing a pure 
Python drop-in replacement for cmath, based on Kahan's "branch cuts for 
elementary functions" paper.  This fixes a variety of problems in cmath, 
including the buggy branch cuts for asinh.  File attached, in case it's 
of any use.

As Tim Peters pointed out, the real problem here is a lack of decent 
unit tests for these functions.  I have tests for the file above, but 
they assume IEEE754 floats, which is probably not an acceptable 
assumption in general.