Is there a way to show the branch in git somewhere visually (background or similiar) in ConEmu?
Yes there is a way. Install git bash, then in ConEmu settings, under the "ComSpec" section set the Explicit executable to "C:\Program Files (x86)\Git\bin\sh.exe" --login -i
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This runs a bash shell session, and gives you a fully resizable window, with the git tab completion and current working branch prompt.