what is the best way to interpolate a set of points in 2d, such that there is only one parameter to indicate position on the curve (like is the case for a Bézier curve)?
one thing I know is that we can use bezier cubic splines and then merge these segments into one Bézier curve, which is an approximation and has limitations in terms of the order of the curve. Is there something similar that gives complete interpolation with an arbitrarily high order curve other than Bézier curve?
Suppose your points have coordinates $(x_i,y_i)$ for $i=1,2,\ldots,n$. Assign a parameter value $t_i$ to the $i$-th point. There are many ways to do this. The usual method is to make the parameter increments proportional to the distances between points.
Now use Lagrange interpolation to construct a polynomial function $x(t)$ such that $x(t_i) = x_i$. Similarly, construct a polynomial function $y(t)$ such that $y(t_i) = y_i$.
Then the curve $C(t) = (x(t), y(t))$ does what you want.
Note that this may not work very well if you have a large number of points (more than around 10). The number of points will dictate the degrees of the two polynomials, and interpolation with high-degree polynomials has several problems. For example, your curves might wiggle in unexpected ways.
You asked for the “best” way. No-one can tell you that unless you describe what makes one curve better than another (in your view).