I don't like search suggestions even in Chrome. I find it makes the omnibar less useful.

The omnibar was the game changer for me when Chrome did it. I never have many tabs open as I can recall anything I've visited previously by typing a few words. More often than not what I want is one of the first suggestions.

So to help with this, I turn off search suggestions since they are not relevant to my past browser activity. I also got in the habit of frequently using Private window so that my History has more relevant content. It's great and I'm confident I don't need to bookmark a lot.