Wordpress - Can you run wordpress with multiple domains at the same time (not multisite)?

Are you wanting for the exact same WordPress site to work on two different domain names, where all links and content would use either domain name?

If this were the case, you would run into Duplicate Content issues with Google, hurting your SEO pretty badly.

That being said, you still would have a very hard time getting WordPress to properly load your content off of either domain name. A variety of functionality uses the get_site_url function to construct the links on your pages, so all of your links would be using only the one domain within your Siteurl setting.

Even still, if you want to try and get really tricksy, you can use something like the following to define your Siteurl and Home in the wp-config.php based upon the Domain being passed to the site:

define('WP_SITEURL', 'http://' . $_SERVER['HTTP_HOST'] . '/path/to/wordpress');
define('WP_HOME', 'http://' . $_SERVER['HTTP_HOST'] . '/path/to/wordpress');