Hi Caroline,
It's generally safe to take vitamins and minerals with TKI's, but usually if you have a deficiency that needs to be corrected. Herbs are more case by case as you noted with St. John's wort.
I also developed low iron because Sprycel suppressed my red blood cell count. I take a heme iron supplement from time to time and that corrected my hemoglobin levels back to normal. I have taken co-Q10 and seaweed, but don't much anymore. I have simply modified my diet to make sure I am eating for nutrition and not eating for fun (i.e. no more sugar). I drink lots of tea, mostly jasmine and earl grey (from Harrods).
The reason you see magnesium, D3 mentioned is because in the West, we have depleted our soils and put on too many clothes and sunscreen. Many people are very deficient in Mg and D (D3 converts to D in the body) and don't know it. Of anything you can do - adding Mg and D3 will do more for your health than just about anything else. Get tested for vitamin D to know your baseline. More is not better with D3. Curcumin is promising, but not conclusive. I take it it in relatively large quantities because I did notice a correlation with my PCR resutls. It is anecdotal, not scientific. What is scientific is that my joint pain is gone. I feel I take Curcumin more for overall well being (inflammation) than I do for CML. Curcumin is good for you. It's part of the Tumeric root and like ginger, beets, onions, and other roots - very good to eat.
I stay away from multi-vitamins. Our bodies don't process vitamins in compressed combined form. Most of it is just wasted.
Hope this helps.