My personal rule of thumb for simplifying a website's chrome (header, footer, navigational items) is do it whenever it makes sense. Amazon was one of the first to do it because they understand that conversions increase the less chances users have to exit the process with other links.
Bred out of necessity, simplifying or even header a website's chrome became fairly popular within mobile apps. The Pinterest iOS app is a great example. They hide the top bar and bottom navigation bar as a user scrolls through content, revealing it again when a user moves up stops moving for a period of time.
Within the last few months, this design pattern has started resurfacing within desktop websites. MailChimp does this (Example: Features) with their new website. Apple's new website also employs this within their new iPhone websites, hiding the overall site navigation as you start scrolling down the page, leaving only pertinent links and focusing on the content.
In all of the above cases, content was removed that could hinder users from completing the primary task. In your case, I would look at ways to start focusing the user as soon as they choose a product. At that point you know they're interested and want to convert them as quickly, easily and effortlessly as possible.