I would wake up to my first alarm, shut it off and close my eyes and finish my dream. I would do this 3 times then I'd finally get up for work. Its not a lucid dream. I can't control what happens, but its like living another life.
Scientific analysis of dreams has yet to uncover conclusive profound insights into their purpose, function and meaning.
I have experienced these phenomena; my guess is that whatever pseudorandom brain noise is in charge of interpreting dreams into a story, does not conduct a full wipe of the signals & memory upon waking, particularly on a sudden waking as with an alarm. It's possible to shut off the alarm mechanically as one gets accustomed to the motion, thus not achieving a fully conscious state - sleep continues unhindered. Or, without new thoughts or elaborate movement rearranging the current state severely, you might take the dream in a similar direction - they're usually based on recent experiences anyway, to some extent. I've had such cases as waking, recalling the details, then changing a certain illogical or undesirable part.
And not sure to what extent goes the separation of lucidity, but 1st person view of a reality that is quite but not entirely unlike your own seems a rather common theme. Recurring subjects may be of greater interest.