Are you shutting down all non-essential programs (especially anti-virus) before starting the game? Firewalls have caused conflicts (generally on loading screens) and overlays from graphics tweaking/monitoring programs or chat programs have also caused issues. This seems more like a driver / DirectX related issue, though.
Check for updated graphics drivers and Windows updates in general.
Try verifying local files, if you are using the Galaxy client; select Divinity: Original Sin Enhanced Edition in the left column, then click on the More button, and in the Manage menu select 'Verify / Repair'. If you downloaded the installer, you can try a clean re-install of the game.
Try temporarily lowering your desktop resolution, to limit what the game will auto-detect.
Alternately, you can edit a configuration file to manually set the resolution and display mode (see below).
Try browsing to the '..\Documents\Larian Studios' folder and deleting the 'Divinity Original Sin Enhanced Edition' subfolder.
This folder contains the saved games, configuration files and a level cache folder. Normally I would suggest renaming the folder for troubleshooting purposes, but if you don't have any saved games that isn't an issue.
Next, extract the replacement folder from
this download into your '..\Documents\Larian Studios' folder, and see if that will get the game to show up.
The graphicSettings.lsx file is set to 1280x720 Windowed mode, Very Low quality preset, which you can change in the options (manually, or hit autodetect) if this helps. If not, delete the downloaded My Documents D:OS folder and try starting the game again.
To manually change the display mode, open the the graphicSettings.lsx file (in the '..\Documents\Larian Studios\Divinity Original Sin Enhanced Edition' folder) in Wordpad, or other text editor, and search for 'fake' near the beginning. Edit the number in value="0" / value="1" in the second line after FakeFullscreenEnabled / Fullscreen; if both FakeFullscreen and Fullscreen are set to zero, it will be in Window mode.
<node id="ConfigEntry">
<attribute id="MapKey" value="
FakeFullscreenEnabled" type="22" />
<attribute id="Type" value="0" type="5" />
<attribute id="Value"
value="1" type="4" />
</node>
<node id="ConfigEntry">
<attribute id="MapKey" value="FrameCapFPS" type="22" />
<attribute id="Type" value="0" type="5" />
<attribute id="Value" value="60" type="4" />
</node>
<node id="ConfigEntry">
<attribute id="MapKey" value="
Fullscreen" type="22" />
<attribute id="Type" value="0" type="5" />
<attribute id="Value"
value="0" type="4" />
</node>
Search for 'Height' to change the resolution:
</node>
<node id="ConfigEntry">
<attribute id="MapKey" value="
ScreenHeight" type="22" />
<attribute id="Type" value="0" type="5" />
<attribute id="Value" value="
1600" type="4" />
</node>
<node id="ConfigEntry">
<attribute id="MapKey" value="
ScreenWidth" type="22" />
<attribute id="Type" value="0" type="5" />
<attribute id="Value" value="
2560" type="4" />
</node>
