Occasionally I run across sheet music that has the left and right hand playing the same note at the same time. Is this on purpose? If so... Why? And how do I play it? For example, see the first measure below. Both left and right hand playing g at the same time.
Usually this happens when the piece was originally for some other instrument, and most likely for more than one. For instance, this might have been an orchestral piece or a song for voices, where it would make sense for two different voices to play (or sing) the same note.
Of course, on the piano you cannot play the same note twice. So, you will only play it with one hand; play the one G and ignore the other.
This is apparently a rather carelessly produced version, since the composer's name is spelled wrongly! She is Imogen Heap, not Imagen Heep. So it's probably not worth trying to guess why the music is written the way it is.
But this notation does sometimes occur in keyboard versions of music that is in several melodic "voices," for example a piece that was originally written for a choir or a group of monophonic instruments. In the original, two different "voices" may sometimes "collide" and sing or play the same note, and that is reproduced in the piano arrangement.
On a piano version, you just play the "doubled" note once, with whichever hand seems easiest.
Of course if you are playing a keyboard instrument with more than one keyboard (organ, synths, etc) there is no reason why you shouldn't play the same pitch note on different keyboards at the same time, and often the two keyboards will be producing different sounds from each other.
This is actually extremely common in keyboard music with any degree of polyphony baked in. I don't think I've written a keyboard piece that doesn't have at least one unison in it.
Yes, you normally play the unison with one finger on one key, with rare exceptions in pieces specifically written for instruments with 2 or more manuals (e.g., Goldberg Variations). The unisons are there to make the voice leading clear, because understanding how the voices proceed will affect your phrasing.
I'd recommend looking over the answers and commentary to Same note to be played with both hands? - this question has come up before.
This is the correct notation of this song:
She’s singing/playing with a vocoder.
I agree with user 16935 that this kind of notation is quite usual. I remember several Bartok pieces training exactly this problem to play the same piano key with both hands, using both thumbs on the little G-key.
We can compare this situation with a picture of a drawing with transparent silhouettes.