I use to play with the iterated sequence $$z_0=0+i0, \quad c=c_1+ic_2,\qquad \quad z_{n+1}=z_n^2+c,$$ and drawing the approximation to the Mandelbrot set, in which $|z_n|\leq r$, to some $r>0$. Here you find how to draw some things with Phyton.

If you write $z=x+iy$, you can see that $$z_n^2+c=(x_n^2-y_n^2+c_1)+i(2x_ny_n+c_2).$$

This gives you a glimpse to define the function $$f(x,y)=(x^2-y^2,2xy)=(f_1(x,y),f_2(x,y))$$ and play with things like $$f(f(x,y))+(x,y)=(f_1(x,y)^2-f_2(x,y)^2+x,2f_1(x,y)f_2(x,y)+y),$$ that can be written as $$g_{n+1}(x,y)=f(g_{n}(x,y))+(x,y),$$ with $$g_1(x,y)=f(x,y).$$ Then you can draw the implicitly curve $|g_n(x,y)|=r$ as you can see here in Desmos.

With the implicit given curves in mind you can use your preferred programing language to draw the curves and animate it.

You can see here the second part of the video you mention in your question.

An interesting discussion on how to parametrize the boundary of the Mandelbrot set that I found on SearchOnMath is this thread.