There is a closed form solution for finding the minimum number of moves the chess knight needs to move a specified displacement on the infinite chess board.  Let $g$ be the requisite displacement expressed as a Gaussian integer; the real part of $g$ will be the horizontal displacement, and the imaginary part of $g$ will be the vertical displacement.  Then we may write

$$g = ((1-i)g+(2-i)d)(2+i) - (g+(2+i)d)(2-i). $$

 Here, $d$ can be any Gaussian integer; the value of $d$ which minimizes the number of knight moves is 
$d=Cint((2i-5)g/10)$, where $Cint$ is the closest Gaussian integer of the argument.  The first term yields the requisite counterclockwise moves, and the second term yields the requisite clockwise moves.  The real and imaginary parts of the counterclockwise and clockwise coefficients together yield the total minimum requisite moves of the chess knight, while simultaneously specifying all minimal paths.