Use this release, not 1.0.0-alpha1. It has the same code. While this project was a submodule of the Design System API, packages.drupal.org published each Design System API release from 1.0.0-alpha1 to 1.0.0-alpha7 as a metapackage named drupal/design_system_api_chrome, which installs no code of its own. This release is numbered past them, so composer require drupal/design_system_api_chrome:^1.0@alpha installs the project.
The first release: core's admin chrome on the Design System API's contract. Navigation (sidebar and top bar), contextual links, and the Toolbar with Admin Toolbar take the chrome context's colour, type, spacing, radius, shadow and focus. Up to Design System API 1.0.0-alpha7 this was the module's design_system_api_chrome submodule.
Status: experimental. It follows the Design System API's contract, which will still change.