Extension work to a paper which is currently under review process : Is it good to communicate this? I have submitted a paper for a math journal and it is currently under review for the past 4 months. Meanwhile I have come up with an important work, that directly builds on top of that paper, which I thinks increases the importance of current paper, when looked in the light of this new work. I can say that this new piece can be a good extention work to my paper increasing its value.

My question is, can I write to the Editor in chief, who has been communicating with me, about this new development. If so, would it speed up or have positive impact on the review process of the currently sumbmitted paper? Or does it also mean that editor may ask me to take back my current paper, add this new piece of extension and resubmit.

Or Should I submit this new piece of work as a separate paper to same journal, even when the current paper is still under review?

In any case, is it a good idea to communicate this to the editor in chief?
I would suggest putting the new results into a new paper. There is no reason to notify the editor unless you plan to withdraw the current paper.

The current paper will/must stand on its own. If it does, then it will be published and the new results give you a new publication, perhaps. 

But if your current paper is rejected for some reason, then is the time to think about integrating the new results with the old, which really would tend to increase the value. 

I think the outcome is probably better if you wait and continue work, rather than creating a possibly confusing situation for the editor.