Is it possible to exchange messages between the extension and the application?

Question

Google Chrome

How to implement messaging between chrome extension and app within browser? Only by means of api chrome, without server solutions.
For example, on the page, select the text and create a note through the context menu (this is the work of the extension). Notes are stored somewhere. We open the chrome application, all notes ever created are displayed. We continue to work with them.
By app I mean https://developer.chrome.com/apps/about_apps
Extension https://developer.chrome.com/extensions

Answer 1

The solution for me is to open a window of type popup (if closed) and implement all the logic in it.

Answer 2

I watched a trojan that sent data from a script in the context of the page to the extension. He did it through window.postMessage().