Why we need Complex Number ?

Why we need to do double differentitaion ?

Let a complex number z as 2+3i, find complex number if we rotate it to 45^(@).

A complex number z is rotated in anticlockwise direction by an angle alpha and we get z' and if the same complex number z is rotated by an angle alpha in clockwise direction and we get z' then

Why do we need appendicular skeleton ?