Hold your left thumb against the first segment of your left pinky and count 1, move up one segment, count two. Count each of the segments on your left fingers. You’re now at 12. Move your right thumb to the first segment on your right pinky, and then start over on your left. Each time you count 12 on the left, move your right thumb up one. You can now count to 144 on your fingers.

(Note, this is not directly related to the base 12 conversation, I just like to share this method of counting whenever I can.)

I like your method a lot, it’s easier than binary with the fingers (which can get you to 1023 with both hands). But it is more work to understand for a base 10 brain than my current method:

One, two, three, four on your fingers of your right hand as usual, then drop all fingers and add the thumb for five. Then six, seven, eight, nine with the fingers.

Then drop all fingers and thumb, and add a finger on the other hand. That’s ten. Continue. You can get to 99 and very rapidly comprehend the value without any real work. Thumbs are 5s, fingers are 1s, one hand is the 1s column and the other hand is the 10s column.