Real vector plus complex vector

[Neitrino]

If there are three vectors a=(1,0,0) and b=(0,1,0) what vector will be a+ib (i - is a complex)
Is it possible to imagine its direction in real three-space...

Sorry if this is basic question

 

[Phiphy]

Just add each component respectively: a+ib=(1+0i,0+i,0+0i)=(1,i,0).
You can not imagine its direction in real 3-space because complex numbers can not be ordered. But you can imagine a 6-dimensional vector with a definite direction.