Chrome's incognito mode is useful if you don't want your browsing history saved to your account, don't want websites to access your cookies, or if you want to troubleshoot your browser. But it doesn't do much to protect your privacy. Your ISP can see what websites you visit, and services like Twitter can figure out who you are even without cookies.
From Tech Talks:
But Twitter also keeps track of IP address, device type, device ID and browser type and version. Technically, it will be able to use all those factors to link your activity to your account. Facebook goes further and even tracks your activity across other websites when you're not logged in to your account.