Is it this one?

This song has been recorded, sometimes in abridged form, by several artists in the past, including Perry Como, Frank Sinatra and Judy Garland. But in the future – three million years into the future – it is sung by the Cat in 'The End'. Thanks to Trace for sending me the rest of Billy Rose's 1927 lyrics, sung to music by Al Jolson and Dave Dreyer.

Me And My Shadow
Music by Al Jolson and Dave Dreyer
with lyrics by Billy Rose, 1927

Shades of night are falling and I'm lonely
Standing on the corner feeling blue
Sweethearts out for fun, pass me one by one
Guess I'll wind up like I always do, with only

Me and my shadow
Strolling down the avenue
Me and my shadow
Not another soul to tell our troubles to

And when it's twelve o'clock, we climb the stair
We never knock, for nobody's there
Just me and my shadow
All alone and feeling blue

When the sun sets on the far horizon
And the parlour lamps begin to glow
Jim and Jack and John, put their slippers on
They're all set but we're still on the go, so lonely

Me and my shadow
Strolling down the avenue
Me and my shadow
Not another soul to tell our troubles to

And when it's twelve o'clock, we climb the stair
We never knock, for nobody's there
Just me and my shadow
All alone and feeling blue