http://umm.edu/health/medical/altmed...ine-lcarnitine

Since this site says "theoretical" effects...This means to me that
the person using it should have blood work done regularly to see if there are any effects. (which he probably does anyway)

However, this study shows carnitine IMPROVES muscle tone in both hypo and hyper patients.
Carnitine can be low in hyperthyroid patients as well.
http://www.ncbi.nlm.nih.gov/pubmed/15803480

There are only 3 pages of studies on this topic on PubMed and most are highly technical and don't refer to hypo patients specifically.

Carnitine is mentioned for helping PN...but it is not a solution for everyone. Certainly if you are concerned you can not use it.

Similarly... CoQ-10 helps the heart in those HYPERthyroid
http://www.ncbi.nlm.nih.gov/pubmed/6532793

CoQ-10 is made by the body, as is carnitine. CoQ-10 is depleted by statin drugs. So if your husband is not using these drugs to lower cholesterol, CoQ-10 is not really necessary.

From what I see in the lay sites is the word "may" and very small studies. I don't see a medical interest in this topic yet, as it might not be showing up in patients.

Some of the Mayo comments refer to large doses with caution.
I saw one comment that most doses under 3000mg a day are recommended for various CoQ-10 uses.

So CoQ-10 is not a must for anyone not using statin drugs at this time. You can avoid it, if you choose.

Most fish oils are purified these days.
Here is a lab site that tests products and this is the fish oil
page:
https://labdoor.com/rankings/fish-oil

Ask your local pharmacists about which fish oils they would recommend and use.