Change URLs depending on alias

You can use the the WP_HOME directive in your wp-config.php. Just set it dynamically based on $_SERVER['HTTP_HOST'] like the example in the codex.

<?php
define('WP_HOME', 'http://' . $_SERVER['HTTP_HOST']);

You might also want to define WP_SITEURL in the same way. I do something like the above for the wp-content url on one of my sites:

<?php
define('WP_CONTENT_URL', '//' . $_SERVER['HTTP_HOST'] . '/content');

All that said, the SEO in me is screaming that this is a terrible idea — duplicate content and such. Much better to have a single site and redirect the alias to the canonical site.