MATL, 28 27 26 bytes

Inspired by Sanchises' answer to a loosely related challenge. I think the approach is also similar to that in Jonah's answer here.

JO2B1i:"6Mh]htn:Eq*^YpYsXG 

The output is rotated 90 degrees clockwise with respect to the examples in the challenge text. The initial segment is not drawn. More steps than necessary are drawn.

Try it at MATL online!

With input 11 this gives F₁₄ as shown in the challenge (rotated):

Explanation

J % Push imaginary unit, j: (*) O % Push 0: (**) 2B % Push 2 in binary, that is, [1 0]. This is F_1, with 0 and swapped 1 % Push 1. This is F_0, with 1 instead of 0 i % Input n :" % Do n times 6M % Push first input to the latest function that took more than 1 input. % In the first iteration this does nothing; in iteration k>1 it gives % F_{k-1} h % Concatenate F_k and F_{k-1} to produce F_{k+1} ] % End. The top of the stack contains F_{k+1} h % Concatenate 0 (**) with F_{k+1}: (***) t % Duplicate n: % Range [1 2 3 ... L], where L is the length of the above Eq % Times 2, minus 1, element-wise. Gives [1 3 5 ...] * % Multiply with (***), element-wise ^ % Imaginary unit (*) raised to this, element-wise Yp % Cumulative product. This generates the directions: 1, j, -1 or -j Ys % Cumulative sum. This generates the path, starting at 0 XG % Plot as complex numbers 

MATL, 28 27 26 bytes

Inspired by Sanchises' answer to a loosely related challenge. I think the approach is also similar to that in Jonah's answer here.

JO2B1i:"6Mh]htn:Eq*^YpYsXG 

The output is rotated 90 degrees clockwise with respect to the examples in the challenge text. The initial segment is not drawn. More steps than necessary are drawn.

Try it at MATL online!

With input 11 this gives the last image shown in the challenge (rotated):

Explanation

J % Push imaginary unit, j: (*) O % Push 0: (**) 2B % Push 2 in binary, that is, [1 0]. This is F_1, with 0 and swapped 1 % Push 1. This is F_0, with 1 instead of 0 i % Input n :" % Do n times 6M % Push first input to the latest function that took more than 1 input. % In the first iteration this does nothing; in iteration k>1 it gives % F_{k-1} h % Concatenate F_k and F_{k-1} to produce F_{k+1} ] % End. The top of the stack contains F_{k+1} h % Concatenate 0 (**) with F_{k+1}: (***) t % Duplicate n: % Range [1 2 3 ... L], where L is the length of the above Eq % Times 2, minus 1, element-wise. Gives [1 3 5 ...] * % Multiply with (***), element-wise ^ % Imaginary unit (*) raised to this, element-wise Yp % Cumulative product. This generates the directions: 1, j, -1 or -j Ys % Cumulative sum. This generates the path, starting at 0 XG % Plot as complex numbers 

MATL, 28 27 26 bytes

Inspired by Sanchises' answer to a loosely related challenge. I think the approach is also similar to that in Jonah's answer here.

JO2B1i:"6Mh]htn:Eq*^YpYsXG 

The output is rotated 90 degrees clockwise with respect to the examples in the challenge text. The initial segment is not drawn. More steps than necessary are drawn.

Try it at MATL online!

With input 11 this gives F₁₄ as shown in the challenge (rotated):

MATL, 28 27 26 bytes

Inspired by Sanchises' answer to a loosely related challenge. I think the approach is also similar to that in Jonah's answer here.

JO2B1i:"6Mh]htn:Eq*^YpYsXG 

The output is rotated 90 degrees clockwise with respect to the examples in the challenge text. The initial segment is not drawn. More steps than necessary are drawn.

Try it at MATL online!

With input 11 this gives F₁₄ as shown in the challenge (rotated):

Explanation

J % Push imaginary unit, j: (*) O % Push 0: (**) 2B % Push 2 in binary, that is, [1 0]. This is F_1, with 0 and swapped 1 % Push 1. This is F_0, with 1 instead of 0 i % Input n :" % Do n times 6M % Push first input to the latest function that took more than 1 input. % In the first iteration this does nothing; in iteration k>1 it gives % F_{k-1} h % Concatenate F_k and F_{k-1} to produce F_{k+1} ] % End. The top of the stack contains F_{k+1} h % Concatenate 0 (**) with F_{k+1}: (***) t % Duplicate n: % Range [1 2 3 ... L], where L is the length of the above Eq % Times 2, minus 1, element-wise. Gives [1 3 5 ...] * % Multiply with (***), element-wise ^ % Imaginary unit (*) raised to this, element-wise Yp % Cumulative product. This generates the directions: 1, j, -1 or -j Ys % Cumulative sum. This generates the path, starting at 0 XG % Plot as complex numbers 

MATL, 28 27 26 bytes

Inspired by Sanchises' answer to a loosely related challenge. I think the approach is also similar to that in Jonah's answer to this challenge.

JO2B1i:"6Mh]htn:Eq*^YpYsXG 

The output is rotated 90 degrees clockwise with respect to the examples in the challenge. The initial segment is not drawn. More steps than necessary are drawn.

Try it at MATL online!

With input 11 this gives F₁₄ as shown in the challenge (rotated):

MATL, 28 27 26 bytes

Inspired by Sanchises' answer to a loosely related challenge. I think the approach is also similar to that in Jonah's answer here.

JO2B1i:"6Mh]htn:Eq*^YpYsXG 

The output is rotated 90 degrees clockwise with respect to the examples in the challenge text. The initial segment is not drawn. More steps than necessary are drawn.

Try it at MATL online!

With input 11 this gives F₁₄ as shown in the challenge (rotated):