CI: add a code formatter tool

This issue aims to open a discussion on adding a code formatter tool such as clang to keep the code conformant to a standard.

The issue names no files, tests, or entry points; begin by reviewing the repository's CI configuration and current formatting practices. Resolve the discussion about which formatter tool and standard to adopt, then define the CI checks and documentation needed for the formatter to be considered complete.

Written by the indexing model from the issue text.