No you can't but for such cases as this, there is still desktop mode for such cases. Besides, there are no modern apps like that, so you have to rely on desktop programs. I managed to get 3 modern apps running side-by-side, one of those could be dekstop on which you could arrange programs even further.
3 modern apps:
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2 modern + dekstop:
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2 modern + desktop split further, you could split it even more, like you did already on your screenshot before.
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No I don't use this feature often on my desktop because I have two screen, but I do snap 2 desktop windows on each screen, sometimes email (modern app) is one of them. You can use Windows 8 without even touching modern apps the same way you did before on 7, or you could take advantage of modern apps (pros for those: writing an app is simpler, distribution is much simpler (you submit it to the store, no need to set up your own servers), you can take advantage of Azure to sync settings across devices - all those things are built in the SDK and had to be taken care of by developer before, including the whole infrastructure behind it).
I don't use many modern apps except calendar and email, those I use regularly. Desktop isn't going anywhere, they are even further improving in the summer as stated by M$ on their Build in April, it will allow you to run modern apps in windowed mode and then it will be possible to arrange them in the way you want to (like in your screenshot).
A word of advice though, invest in a second screen if you multitask so much. You can get a decent 1080p for 150$ and it makes your life much easier.