Question 679034
The conditions needed are:

1)

{{{a[1]^2+a[2]^2=1}}}

2)
{{{b[1]^2+b[2]^2=1}}}



3)
{{{(a[1]+b[1])^2+(a[2]+b[2])^2=1}}}


This nonlinear system does have solutions, for example

{{{a=1i}}}
{{{b=-1/2 *i + sqrt(3)/2 *j  }}}

is such a pair