Is there a way to show the branch in git somewhere visually (background or similiar) in ConEmu?
Disclaimer #1
ConEmu is not a shell, so it does not provide "shell features" like tab-completion, command history and others.
Usually, Git information like branch or amount of changes is displayed in the command line prompt, for example C:\path\to\repository [branch|+2~4-6]>. This is done by the executing shell, not the console frame.
As far as I know, the standard Windows command prompt (CMD) does not support modifying that. The bash that comes with Git for Windows already supports that perfectly though; and for PowerShell there are numerous extensions for Git, most notably posh-git.
Git branch can be visible in plain cmd or Far Manager prompt.

All magic is done with special ANSI sequences ("Inject ConEmuHk" and "ANSI X3.64 ..." options must be checked).
I
Run GitShowBranch /i to install showing branch, GitShowBranch /u to uninstall.
Also, you may run your cmd as following (within Task contents or ConEmu's Command line)
cmd /k ver & GitShowBranch /i
PS. File GitShowBranch exists in ConEmu's distro, but you may see it online.
Change the specified named task to {Bash::Git} in Settings -> Startup and you'll have branch name showing up.

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.
This runs a bash shell session, and gives you a fully resizable window, with the git tab completion and current working branch prompt.
To sum up the situation is:
I know sh.exe can do this but needed to check if it is in the Git\bin folder and have access by only writing sh in command line.
First I saw sh.exe in Git\bin folder but wanted to see if I can execute on command line. To see it I checked environment vars using by echo %path%
It was complicated to see if there is. I used powershell script and there is :)
I would have execute sh in command line too :)))
Refrences:
Codes:
echo %path%
($env:Path).split(";") | where ({$_ -like "*Git*"})