You can set your chrome extension to function in incognito mode (manually the is no option for automatically set it through code or by default)

Image from Chromium Blog.
Then to check if it is working in incognito mode :
chrome.extension.isAllowedIncognitoAccess(function callback)
Retrieves the state of the extension's access to Incognito-mode (as determined by the user-controlled 'Allowed in Incognito' checkbox.
Code Sample by Rob
chrome.extension.isAllowedIncognitoAccess(function(isAllowedAccess) {
if (isAllowedAccess) return; // Great, we've got access
// alert for a quick demonstration, please create your own user-friendly UI
alert('Please allow incognito mode in the following screen.');
chrome.tabs.create({
url: 'chrome://extensions/?id=' + chrome.runtime.id
});
});
Hope this helps!