Does your hair stop growing when you die?

No, it does not.

Hair itself is not alive in any case. The hair follicle, in the skin, is living tissue but that's the only part or hair that is.

After death, however, the body dehydrates fairly quickly, and this dehydration causes the skin to shrink and tighten. This may create the illusion of hair growth (and nail growth), but both are an illusion.