Stringify DOMWindow object

For some reason I seem unable to use JSON.stringify on a DOMWindow object. For example:

console.log(window.self); // Outputs a hierarchical DOMWindow object
console.log(JSON.stringify(window.self)); // Outputs nothing - not even an error

alert(window.self); // Alerts "[object DOMWindow]"
alert(JSON.stringify(window.self)); // Again nothing - not even an error

Tested on Safari and Chrome. Does anyone have any ideas how I can achieve this?

Edit:

Moved edit to a new question as it's not really specific to this.