monitoring history.pushstate from a chrome extension

Not sure whether you trying to do this in background.js or content.js, but if it is the former, you can do so using webNavigation events:

You need to set permissions for webNavigation in manifest.json:

  "permissions": [
     "webNavigation"
  ],

Then in background.js:

  chrome.webNavigation.onHistoryStateUpdated.addListener(function(details) {
        console.log('Page uses History API and we heard a pushSate/replaceState.');
        // do your thing
  });

Source: Chrome Extension docs for webNavigation

Yes,

One way is to create a script tag and put your code there:

html = "window.history.pushState = function(a,b,c) { alert('Change !'); };";

var headID = document.getElementsByTagName("head")[0];         
var newScript = document.createElement('script');
newScript.type = 'text/javascript';
newScript.innerHTML = html;
headID.appendChild(newScript);