my oldest quit napping at 18 months and was fine all day and slept well through the night. He still doesn't require a lot of sleep as a teenager. I would still have those few hours of quiet time though and kept his normal schedule but instead of his nap, I would put up the baby gate to his room with the door open so I could still hear him and check on him. He would play with his toys in his room and sometimes would fall asleep but didn't always fall asleep. He learned to be independent with having those few hours of alone time and he never fussed about having to be in his room for naptime but I didn't care if he went to sleep or not as long as he had his quiet playtime and I could get some housework done as that was usually my time to get things done. He loved to look at books and by age 2 he loved to do puzzles. He has always been a very fast learner. by the time he was 3 I didn't even put him in his room for those hours, he would sit at the table and do 100 piece puzzles or write his letters on his magna-doodle for hours and would keep himself content as I had a colicky baby when he was 3 and he did very well with being patient and playing by himself while I was dealing with her fussing everyday for 5 1/2 months. I did take special time for him in the mornings when she wasn't fussing and do things he wanted to do like read him a book, help me cook, play outside, or even watch a movie together.
my daughter took naps until she was 4 years old. She seemed to need them and would get cranky if she didn't get them.
my youngest son took naps until he was about 2 1/2.
They all slept through the night by 8 weeks and would usually go to bed at 8:30 or 9 and wake up in the morning around 8:30.
Every child is different and all of my kids liked to have their independent time to themselves so they didn't mind playing in their rooms by themselves when they weren't sleepy for a nap. When they were babies and would wake up in the mornings in their cribs, I would let them play for 15-20 min before I went to get them. I had a baby monitor and loved hearing them coo and play in the mornings. They would talk to things on the wall or the toys hanging on the crib. I think it helped them be more independent and they never did hit that clingy stage where they had to be in the same room with me all the time or hang on my leg when cooking dinner. I have seen other babies do this at about a year and the parents have a hard time leaving them with anyone or getting anything done because the baby requires so much attention.