This question was inspired by At What age does a baby recognize themself in a mirror? and the memory it brought up for me of the day we were on a family "hike" at a local botanical garden and my daughter (age two?) looked down and started flapping her arms. You could see that she was testing if the gray thing at her feet would also flap and when it did, she giggled with glee and started "running" in that cute almost falling way kids do while flapping her arms and giggling at the fact that her shadow was also flapping away.
Obviously my daughter was walking at this point, but I have video of her discovering her reflection at an age when she was "almost" walking but not quite. I wondered if the discovery generally happens in the same range of time or if they are two distinct "discoveries" in terms of cognitive development.