Surprise there is no 'normal.'. What bothers one person with eczema can be a life-saver for another.
I have eczema and am allergic to soap, all cosmetic moisturisers and even E45. I am cautious with DD because of this even though she isn't too bad with eczema.
We've only ever bathed her with Aveeno. You can use something cheap like aqueous cream though but I find it clogs up the plughole a bit as it's thick.
As I said, I used an Aveeno cream on her face but I felt it did no good (maybe even made it worse.) I tried Diprobase and it works well so I have stuck with that.
I use sudacrem if anything ever looks very sore as an emergency and then use diprobase for day to day. I tried aqueous cream and it made no difference.
I haven't bothered with any cosmetic type moisturisers as I have learnt from my own experience that they are often expensive and have alcohol/fragrances or additives which provoke a reaction.
Basically, Surprise if you've found a way to manage it, stick with it. If you need to try new things I would avoid the sorts of things you buy in toiletry departments and go for the pharmacy stuff which is more 'pure.' As PP said, you could try olive oil (I didn't like the grease).
OP, if you don't see an improvement (a lot of babies have dry skin at first) and recommended creams don't work you could consider dairy, especially if you or Dad has eczema or asthma.
My DD had a milk allergy at birth (which often goes with eczema). I cut out dairy as I was BFing and I believe this helped with her skin as well as digestion. Now that she is older and can tolerate cow's milk we both have dairy and her skin doesn't seem affected.