Orthonormal basis for subsets of C^3

gysush
We want to find a basis for W and W_perpendicular for W=span({(i,0,1)}) =Span({w1}) in C^3

a vector x =(a,b,c) in W_perp satisfies <w1,x> = 0 => ai + c = 0 => c=-ai
Thus a vector x in W_perp is x = (a,b,-ai)

So an orthonormal basis in W would be simply w1/norm(w1) ...but the norm(w1)=0 (i^2 + 1 = 0)

What am I missing here? Does a basis for W satisfy that it has zero length? Thus it is just the origin. Then would all of C^3 be W_perp?