Once a respectable English word, deriving from Old English /elles hwat/. Unfortunately it died out.
See also elsewhere, and compare elsewhen (1418), elsewho (1542), elsehow (modern).
It takes time and effort which might usefully be spent doing elsewhat. (The Face, Nick Logan)
His mind was on elsewhat that hot summer evening after tea. (Memories of the Gorbals)
But his eyes were saying elsewhat (One shining summer, Quinn Wilder)
like elsewhere but applying to:
What have you been doing the six weeks while I was away, skydiving?
Naah.. I’ve been busy with elsewhat lately...