Modern as in.. from what year onward ? Personally, I would still choose Half-life 2. I played it right after release and it was amazing at that time. The story was great, gameplay was good, the graphics were really nice, the soundtrack was very well suited for the environment. Great game for me. I even played it a few months ago when that graphical mod/update came out and it was still a very good game for me, even after so many years.
Difficult to decide. As you mentioned, the obvious answer would be Skyrim. When it comes to the vanilla version, there are some interesting quest lines, the dynamic leveling system (while can be exploited) is pretty good, the world is huge but what makes it the "best" is the modding community.
You can make Skyrim an entirely different game if you choose to do so. So from that perspective alone, there is huge variety on what you can do with it. You can make the combat more exciting, make the game more difficult, fix some A.I. issues, populate the areas with more NPCs, roam the world in a car, ride dragons, use guns etc, the possibilities are endless.
There may be games that are better story-wise but I haven't played much of them (e.g. Pillars of Eternity).
I can't think of any other relatively recent PC game that allows you to change that much about your gameplay experience, so yeah, I'll go with Skyrim for now.
For me, Witcher 3 instantly became my favorite PC game and it took the gaming world by storm. The graphics are beautiful, great gameplay and an interesting story-line. There's so much customization in the game, but also has the feel of a great story game. If you haven't tried it yet, invest the money on whatever console and get it. I got it for PC and happy I did because I have a comp that's able to support the graphics on high.
Skyrim is another good choice, but I like the feel and gameplay of Witcher 3 better.