I just took some antidoron home and shared it with those who couldnt/didn't attend?
That's the primary reason of taking prosphora home.
However, indeed, in a monastery I've seen the practice RaphaCam has described, once my priest has also advised me to do it.
So, that after the morning prayers, having not drunk nor eater before them, you take a piece of prosphora and it and, and drink the holy water.
Once I've been also reminded in Romania by a church lady
(when my friend and I took a piece of prosphora in front of the iconostasis) that you can't take it without fasting before it; but firstly she was concerned if my friend and I are Orthodox (both we are)