Thank you for your response. It seems to be working now.
This keeps happening: I create a page, add content to it, try to translate and it does not work: 404. I tried over space of two hours, then I created a blank page that was broken as I described in my request, so that I can reach out to WPML support.
Few hours later it looks like issue was never there.
This is so frustrating, I do not create tickets for fun.
I just tried to translate the original page (hidden link) again and it worked, no problem. I tried to translate this page for two hours and it was broken.