Sorry, maybe there is some difference between blender’s shrink/fatten command and the c4d normal move command that I’m not aware of. I haven’t used c4d before, so I have no way to compare the two operations.
When I want to make something “fatter” I select all of the vertices in that object and use the alt + s command which gives results like this:
Here is the result on the object you linked to:
And the active tool version of this command has this icon:
Now, lets say I wanted to make the ear in this monkey head larger. In that case, instead of using alt + s on the ear portion, I’d snap the cursor to the selection for the ear, then adjust it’s position along the x axis by holding shift and dragging that property slider in the sidebar. Then I’d use the cursor as a pivot point and use a regular scale command (S key):
Edit: I have a hard time visualizing how moving each face out in the direction of their normal could work differently from the alt + s command. Could you share a link to a video that shows this tool in use, or a screenshot of that handle after using the move along normal command in c4d?