I was considering getting a new PC for Fallout 4 but I decided to try it on my current PC (a tad below minimum recommendations) first to see how it worked. Honestly, I'm mostly satisfied with the results and don't feel the need to buy a new PC yet.
I've got an AMD Phenom II X4 940 with a Radeon HD 7850. The graphics card was upgraded a few years ago when my old one burned up. It set me up on medium and it's performed well except a few particular instances where it flatlined to "0 fps". Searchlights, for some reason, seem to be the biggest problem, and possibly waves on large bodies of water. I turned some unneeded effects off and it helped tremendously. I am satisfied.
I'm sure I'll get another PC within the next couple of years, though, or maybe even the next time I see a great deal. Right now, I'm struggling between processors. AMD 8300 series or should I pay the extra for an i5? Do I need an i5? I don't care at all about Ultra settings, by the way. I just want stable frames per second.