Go for print composer. Increase the page size and resolution. Add the map and if you want to create a map of the whole world, make sure you zoom to full extent. Then save as image. If the size is not big enough, increase the paper size further. A0 with 300dpi should easily be possible if you have enough RAM.
As @Dave Pitman points out in his answer, that a user, Stu Smith, got the manifest hack to work.
I actually got it to work too. Follow the instructions using the link:
http://www.danantonielli.com/adobe-app-scaling-on-high-dpi-displays-fix/
Once the manifest.txt file is downloaded, copy it to the QGIS bin folder. In my case:
C:\Program Files\QGIS 2.18\bin
and rename it to 'qgis-bin.exe.manifest'.
Additionally, you'll need to ensure display scaling is disabled.
To do this, you'll need a shortcut to 'QGIS Desktop', for example, by pinning it to the Windows taskbar.
Right click the QGIS icon, then right click 'QGIS Desktop 2.18.3' in my case, and select 'Properties'.
Click the 'Compatibility' tab, and uncheck 'Disable display scaling on high DPI settings'.
There's a minor niggle that the text and icons are not as sharp as they would be otherwise, apart from that the hack works pretty well.
Update:
For the Windows 10 Creators update, in the 'Compatibility' tab, check 'Override high DPI scaling behaviour. Scaling performed by:', and then select 'System' from the drop-down menu.
Best Answer
Find the qgis executable. If installed with OSGeo installer, it is in C:\OSGeo4W64\bin.
Right click on qgis-bin.exe and choose Properties.
Select the 'Compatibility' tab (the second tab).
Select 'Change high DPI settings' at the bottom of the tab.
Check 'Override high DPI scaling behavior' at the bottom.
Select 'System (Enhanced)' from the drop down.
Now, Start the application.
https://danantonielli.com/app-scaling-on-high-dpi-displays-fix-2019/