Google tries to personalize the results as much as possible, they use cookies and other types of identification data to try and track users and provide the most relevant results.
After you close chrome technically there is no method for Google to identify who you are, hence you may see different results.
Getting accurate search position results for tracking marketing progress seems a bit of a buzz word at the moment. It comes back to as Marketers/SEO's we need to track and monitor our efforts, on the flipside Google is trying to provide the most accurate result to the user often by means of personalisation. That is where you have an immediate conflict, for maximum user engagement (which all of us want) you sometimes have to sacrifice tractability.
Assuming you did manage to track personalised results you would need a lot of data about that person and their browsing habits to understand how you back best matched their search query. Most of us would quickly get ourselves lost in data. Does that make sense?
There are plenty of Serps checker tools around, but I sometimes have my doubts how accurate they are - Google definitely doesn't like them.