So, after writing a small programm that sends and receives data from the serial monitor, I tryed it with the Arduino connected to one end of the wireless system and my computer to the other end. It does not work 
The computer does not see the Arduino. The computer (Mac book) does see the wireless system but it seems to only support keyboards and similar devices (my Powermate does work). Even after connecting my keyboard to the wireless system and plugging the Arduino into the USB hub of the keyboard, the Arduino is not seen by the computer, also the keyboard is working.
Thats quite a bit disappointing. I will have a look into setting up the Arduino as a "keyboard" (there is a USB keylogger example somewhere). But I don't know if that will work. I will also look into other wireless HDMI transmitters and the wireless shield for the Arduino (but that might be problematic with other wireless systems on set).
Any ideas about wireless connections between Arduino boards?