4 month old infant with eczema on back of neck and bleeding due to scratching. How would you treat this?

See your doc. There are multiple medications that can be used. First you have to rule out secondary bacterial infection. See your physician.
Depends. Atopic dermatitis, commonly known as eczema, in young infants often involve forehead, facial cheeks, scalp, and torso. Eczema-like skin condition, i.e., contact dermatitis is also a possibility. I would rather have your infant evaluated by his/her pediatrician.
Beginning eczema. Eczema on any part of the body at a young age puts the infant at risk for local skin problems, eye problems and systemic immune problems. You need to see a pediatrician for this who can prescribe the proper medications. A four month infant scratching to the bleeding point should not be neglected long for this problem, .