I have a Hackintosh, iMessage working no problem under 10.8, after upgrade to 10.9, iMessage stop working and pops up "contact apple for support" window.
I tried every possible method under this post(incl. any alternative solutions in other posts). Non of them work for me .
1. iMessage fix with fileNAVRAM ->doesn't work for me, even install fresh system for several times.
2.Add credit card -> my id is associated with CC, no good luck
3.Have a real apple product -> I do have apple product, tried sign in sign out many times, no good luck.
4.Change password -> did but no good luck.
the last thing i can do is to call apple...and it DID work for me. (you will need a real apple product because it will ask you to provide this number)
If you have the similar situation like me, maybe worth time call apple toll-free support number..
I wish i could do it earlier rather spend 10+ hours try to fix this problem by myself.