R, 205 153 147 bytes
-52 from Dominic van Essen.
-4 from Giuseppe.
function(s){n=nchar(s)
m=matrix(" ",2*n,2*n)
for(x in 1:n){m[Re(F)+n,Im(F)+n]=substr(s,x,x)
F=F+c(0,0,2+1i,2-1i)[(-2*x^.5)%%4+1]-1}
write(m,1,2*n)}
R,
att
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