Rod of Asclepius, symbol of medicine used by medical organizations.
An image trademark is often a logo. There are many common symbols that can be part of a trademark. Among those symbols are Caduceus, the twin serpents intertwined, which is often topped with a set of wings, and the Rod of Asclepius, the single serpent wrapped around a pole, or rod.
The Rod of Asclepius is considered a legitimate symbol of medicine and is used by respected medical organizations and schools. The use of Caduceus as a medical symbol is considered a historical mistake made by the US Army and carried on today by groups such as insurance companies and some sellers of medical-related goods.
If the product or service for which you are seeking a trademark is not legitimately medical, if you use either of these symbols, it is likely to be rejected by the Trademark Examiner as "misleading."
Always make sure you know the meaning of any symbols you plan to use in a trademark and also know how these are treated by the Trademark Office.
Another medical symbol, the Red Cross, is protected by the Geneva Convention and by law and is for use in the US only by the American Red Cross organization. This symbol cannot be used in any trademark, except by Johnson & Johnson, which was already using it. You can read more about the protections of the red cross symbol on this downloadable flyer: https://www.redcross.org/content/dam/redcross/Brand-Creative/153701-07-OGC-Trademarks-Flyer.pdf
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When you are designing a logo or other trademark, be sure to check that the logo or design uses proper symbology. It is important to understand what each element of the design means. It is also important to understand how the symbols are treated by the US Trademark Office. For example, a trademark cannot use the complete flag or coat of arms of the U.S., any state, municipality, or foreign nation. Even if you add words or other images around the flag, you cannot use it as a trademark.
Use of a flag in a trademark might be allowed if it is heavily stylized. Flags that are historic and no longer in use, fantasy flags, etc., might be allowed.