I'm looking for (hardware/software) a possibility to have a screenshot
shown on a monitor, without a PC. So it should probably be a connector
between PC and monitorcable which puts this screendump in memory and
keeps this image alive, also when the pc is not connected. Should
especially work with dockingstations

Are you trying to make something appear on a monitor after the PC is turned off?

You're not going to get any physical image(s), but stuff that's built
into the hardware might point you in the right direction. You're not
going to get anything more than 8-bit colors..

http://video-equipment.globalspec.com/LearnMore/Video_Imaging_Equipment/Video_Equipment/Scan_Converters
- some info about converters
You must be a member of this site (globalspec.com)
By some links You can find companies and their production.
p.s. pc must be turn on, it's a source of signal.
[sorry for my english]

" So it should probably be a connector
between PC and monitorcable which puts this screendump in memory and
keeps this image alive"

Between the PC and the monitor cable is a graphics card (video card),
which indeed  contains memory, and circuitry to keep the image alive. 
The graphics card is housed in, powered by, and driven by the PC; it
won't work with the PC turned off. Perhaps if you stated what you
actually want to achieve (e.g., reduce power requirements, prevent
access to the PC or changing the image, etc.), someone could help you
with a solution.