Phew, I’ve bought a new monitor for 140€ + 9€ for a high quality HDMI cable. And it works! I am free again! 
I would have had to try very hard to stay relaxed if the problem turned out to be something other than the monitor 
I’ve got this guy: http://www.samsung.com/de/consumer/displays/displays/home-displays/LS24D300HS/EN