The whole "first dream of the new year" concept is a common phenomenon in Japan, with your first dream said to be prophetic of your whole year, which is rather ironic in this case since all of the dreams are weird, random and hard to figure out the meaning of.
Chiyo says that her removable pigtails that can be used for flying are "very in right now", as if they were a fashion trend.
Once again, Tomo and Yomi do traditional Japanese sketch comedy, including the use of the phrase "Nande ya nen" (meaning something similar to "What do you mean?", "Forget about it" or "What the hell?").
Yukari appearing on the title page in her bed sleeping is rather odd, since her first dream is not shown and she does not even appear for the entire chapter.
Why Chiyo says the reason she is flying is that she is only ten years old is unknown.
Yomi apologizes to Tomo in the latter's dream, though we do not find out why.
The greatest mystery of Azumanga Daioh is probably the very existence of Chiyo-chichi and his appearance in dreams. He doesn't properly tell Sakaki how to "find a real cat" either.
Everything about Osaka's, Tomo's and Sakaki's unconventional New Year's dreams, generally. Tomo's shows what seems to be her greatest desire – namely, being the best at everything. Yomi seems to think of Tomo as superior to herself, Tomo beats Chiyo at studying and also Sakaki at sports. Tomo earlier expressed desire to do so, and, unlikely as it may seem, she beat Chiyo at a P.E. test.
The first appearance of Chiyo-chichi, the strange cat-like creature who claims to be Chiyo's father, is in Sakaki's New Year's dream. He says that later on, Sakaki will find a real cat, and about a year and a half later, she does.
When the girls go back to school, Chiyo appears confused because Sakaki, Osaka and Tomo all had dreams about her, but they won't reveal the details, for different reasons.
Why're you flyin', Chiyo-chan …?
Ehh? Don't you want them? But they're so in right now!
— Chiyo (to Osaka)
Of course! Those things're controllin' her!
— Osaka (on Chiyo's pigtails)
Wahhh! It was so hard, I got a zero!
Plus, I may be fast, but not as fast as a car. There's always someone better than you.
— Tomo (to Sakaki)
It may surprise you to learn that I am not really a cat. I am Chiyo's father. Thank you for watching over my daughter. You must venture forth to seek a real cat. You've got a ways to go before you will, though.
Eh? What's the matter?
— Chiyo (on the girls' reactions to her appearance in their dreams)