The incident was captured on surveillance video.
Natalie Bree Hajek-Richardson says she was in the checkout line at a Nordstrom Rack store when a young child started throwing a tantrum on Monday.
"It didn't bother me that the child was throwing the tantrum, but the volume was very loud; it was hurting my ears," Hajek-Richardson said.
The woman says she asked the child's mother -- politely, she says -- to do something. The two women then got into a verbal altercation in the store.
"She came to the side of me and told me not to tell her child what to do, and I told her that 'I didn't tell your child what to do, I asked you very nicely to calm down your child,'" said Hajek-Richardson. "I told her to go to hell, and she told me I'll see you there."
She says the woman followed her out to the parking lot and punched her twice in the face.
"She was asking me, 'Where'd you tell me to go?' So I repeated again what I said to her, and I told her that I told her to go to hell," Hajek-Richardson said.
Police are looking at the surveillance video to see if they can identify the other woman.