Normalize dots in script names

CHANGES.rst

@@ -12,6 +12,7 @@ Unreleased
 -   Fix return value and type signature of `shell_completion.add_completion_class`
     function. :pr:`2421`
 -   Bash version detection doesn't fail on Windows. :issue:`2461`
+-   Completion works if there is a dot (``.``) in the program name. :issue:`2166`
 
 
 Version 8.1.3

src/click/core.py

@@ -1129,9 +1129,13 @@ def _main_shell_completion(
         :param complete_var: Name of the environment variable that holds
             the completion instruction. Defaults to
             ``_{PROG_NAME}_COMPLETE``.
+
+        .. versionchanged:: 8.2.0
+            Dots (``.``) in ``prog_name`` are replaced with underscores (``_``).
         """
         if complete_var is None:
-            complete_var = f"_{prog_name}_COMPLETE".replace("-", "_").upper()
+            complete_name = prog_name.replace("-", "_").replace(".", "_")
+            complete_var = f"_{complete_name}_COMPLETE".upper()
 
         instruction = os.environ.get(complete_var)