The issue of supplements is murky at best.
Myself, I tend to the bias that a decent diet supplies all the nutrients we need, and that a diet that requires us to take supplements is ipso facto an unhealthy diet. Add to this the fact that much of the research into how much we require of various nutrients was done on a population of carb-eaters, and there are indications (not proof!) that we may do just fine on lower levels of certain nutrients if we are on a ketogenic diet.
The most certain of these is Vitamin C, at least in the case of ketonians who eat meat, since fresh meat has been known for centuries to prevent scurvy (the problem the British Navy faced was that fresh meat was impossible to give their sailors on long deployments; limes and lemons were easier to supply). Not to mention that our need for outside antioxidants appears to be much less on a ketogenic diet.
There are indications that the body uses certain other hormones and nutrients more efficiently on a ketogenic diet, but I wouldn’t claim any of that as actual proof, at this stage.
Vitamin D is interesting, because our body makes it from cholesterol. This means that people who are on a cholesterol-lowering medication or diet may very well need to supplement it. The same is true of testosterone and progesterone, which are also made from Vitamin D.
As for pro-biotics, I don’t think we know nearly enough about our intestinal microbiome to be able to say what works and what doesn’t. I haven’t read anything for or against pro-biotics that I would consider conclusive. I have no idea about whether α-lipoic acid or DHA do any real good, but I strongly doubt they cause harm, either.
Another consideration is that there are data to suggest that nutrients in food are much more effective than the exact same nutrients in pill form, which in turn suggests that the wisest course might be to eat real, whole foods and not worry about supplements. But again, that’s where my bias lies, so read this with a grain of salt.