In the footage, which has been released by police, the woman can be seen calmly walking into Boots on The Strand in central London.
Noticing a one-year-old baby in his buggy nearby, she changed her path to approach him.
The baby's mother, who was looking at something on a shelf when the woman approached, initially thought she was "going to just look and fuss over her baby".
However she was left horrified when the woman grabbed her baby by the throat, and quickly moved push her away from her child.
The woman then allegedly told the child's mother: "That baby would rather be dead."
Police have revealed that - thankfully - the baby was not injured due to the mother's quick reaction, although both mother and child were left shaken up after the incident.
Officers have now released CCTV footage of the horrifying daytime assault in a bid to track down the woman, who is said to be in her early 60s, chubby, with light hair and a boil or deformity on her face.
At the time of the assault, as you may be able to see from the footage, she was wearing a red and white striped poncho with a pair of trousers.
**Anyone with information is asked to call 101 or contact Crimestoppers anonymously on 0800 555 111. **