So I entered Thailand mid Dec, got 60 days, which expires soon. Plan was to extend for 30 more days and then leave.

Now I have a health issue that needs immediate attention, likely an operation in a few weeks and a period of time recovering that will overshoot the runway even with the 30 day extension.

So is it possible to go onto a medical visa and what are the ins and outs of the medical visa?

Medical visa, no. You would have to get that before you traveled to Thailand.

Medical extension, possibly.  Your hospital/doctor must arrange documents that state you CANNOT leave Thailand.  Not that you shouldn't, or that it won't be easy, they need to say that you are unable to travel. With that you might be able to get an extension at immigration for medical reasons.  But these have been notoriously difficult to get, especially in Bangkok.  Basically if you are able to go to the immigration office, in their eyes you are well enough to travel and leave Thailand.  For those who are truly in bad shape the hospital has to arrange the extension for them and those are the situations where it is generally acceptable to immigration. 

Regardless, the first step is going to be your doctor/hospital to see if they've dealt with this before and get their support.

OP, good answer above. 

Here is old thread with couple of good posts outlining process.

 

https://aseannow.com/topic/609357-visa-extension-due-to-illness/

 

@Sheryl may have added advice..