You should stick to one domain for your website so that your customers and search engines know where to find your content. When moving a website across from an old domain, make sure you carefully set up a 301 redirect from each of the old URLs to the new URLs. If you don't, then you're going to cause confusion, lose traffic, reduce your domain authority, increase maintenance costs, and create more work for yourself. If you really need a separate section for your website – perhaps for checkout pages or company blog – consider creating a subdomain instead. For analytics purposes, Google will consider your subdomain a separate site. But users will infer that it's still part of your main website, and, therefore, trust it.