Abstract: Let j:C²´C²®C, j((x₁,x₂), (y₁,y₂))=(x₁- y₁)²+(x₂- y₂)². We say that f:C²®C² preserves distance d³0, if for each X,YÎC² j(X,Y)=d² implies j(f(X),f(Y))=d². We prove that each unit-distance preserving mapping f:C²®C² has a form I°(g,g), where g:C®C is a field homomorphism and I:C²®C² is an affine mapping with orthogonal linear part. We prove an analogous result for mappings from K² to K², where K is a field such that char(K) Ï{2,3,5} and -1 is a square.