Set contents of a field to only display to admin or the user themselves? Yes it's possible, you'll need to create a field and set it to not show on profile. Then using substitutions in a delimiter field you can set only the moderator and user to see the field. Please see tutorials in signature as you'd need to use advanced IF statements. Example as follows.

[code:1]
[cb:if user="#displayedOrMe" OR usertype="Super Administrator"][cb_mysecretfield][/cb:if]
[/code:1]

effectivly the goal is that administrator and the user can exchange information in private on user profil.

Correct, using the above substitution based method will achieve this.

dos that solution is possible with basic community builder or with an special extension.

Yes; delimiter fields, settings fields to not display on profile, and of course substitutions are just some of the many features available with CB out of the box.

However, we do appreciate the support of our subscribers to help further/future development.

But if I don't show a field in user profile, where can I see this feld?

That's what the delimiter field is for and its substitution. Please see tutorials in signature for substitutions or the following example content for your delimiter field.

[code:1]
[cb:if user="#displayedOrMe" OR usertype="Super Administrator"][cb_myeditorfield][/cb:if]
[/code:1]

And this support doc, is it in french?

No, documentation is provided in English only, sorry.