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I suppose that "non-compact complex algebraic curve" means complex affine curve.

The following counterexample was proposed by my friend Fedor Pakovich.

Let $D=\mathbf{C}\backslash\{-1,1\}$. Consider the $4$-th Chebyshev polynomial $$p_1(z)=2(2z^2-1)^2-1=8z^4-8z^2+1.$$ It has critical points at $0,\pm1/\sqrt{2}$, with critical values $\pm1$, therefore it defines an unramified covering $$p_1:C\to D,\quad\mbox{where}\quad C=\mathbf{C}\backslash p_1^{-1}(\{\pm1\}).$$ Now $p_2(z)=-p_1(z)$ is another unramified covering $C\to D$ of the same degree, but evidently $p_1\neq p_2\circ\phi.$

Remarks. 1. This example can be much generalized, of course; one can take any $D$ possessing a non-trivial automorphism $\psi$, then in most cases $p_1$ (mapping whatever Riemann surfce to $D$) and $p_2=\psi\circ p_1$ will not be related as stated in the problem.

The problem will become harder if under the same assumptions we relax the conclusion to $p_2=\psi\circ p_2\circ\phi.$ Fedor and I believe that counterexamples may still exist, but they will be rare.

I suppose that "non-compact complex algebraic curve" means complex affine curve.

The following counterexample was proposed by my friend Fedor Pakovich.

Let $D=\mathbf{C}\backslash\{-1,1\}$. Consider the $4$-th Chebyshev polynomial $$p_1(z)=2(2z^2-1)^2-1=8z^4-8z^2+1.$$ It has critical points at $0,\pm1/\sqrt{2}$, with critical values $\pm1$, therefore it defines an unramified covering $$p_1:C\to D,\quad\mbox{where}\quad C=\mathbf{C}\backslash p_1^{-1}(\{\pm1\}).$$ Now $p_2(z)=-p_1(z)$ is another unramified covering $C\to D$ of the same degree, but evidently $p_1\neq p_2\circ\phi.$ (The only non-trivial automorphism of $C$ is $\phi(z)=-z$).

Remarks. 1. This example can be much generalized, of course; one can take any $D$ possessing a non-trivial automorphism $\psi$, then in most cases $p_1$ (mapping whatever Riemann surfce to $D$) and $p_2=\psi\circ p_1$ will not be related as stated in the problem.

2. The problem will become harder if under the same assumptions we relax the conclusion to $p_2=\psi\circ p_2\circ\phi.$ Fedor and I believe that counterexamples may still exist, but they will be rare.

I suppose that "non-compact complex algabraic curve" means complex affine curve.

Then the answer is negative. The simplest example is this: let $C=D=$ triply punctured sphere, and degree is $6$. (There is only one affine curve isomorphic to the triply punctured sphere). One can construct two rational functions $p_1,p_2$ of degree $6$, each with three critical points and three critical values, such that the local degrees of $p_1$ at the critical points are $(3,5,5)$ and local degrees of $p_2$ are $(4,4,5)$. Evidently there is no isomorphism $\phi$ such that $p_1=p_2\circ\phi$.

Various constructions of such $p_1,p_2$ are available, see, for example

A. Eremenko, A. Gabrielov, M. Shapiro and A. Vainshtein, Rational functions and real Schubert calculus, Proc. AMS, 134, 4 (2005) 949-957.

Alternatively, one can use subgroups of the modular group. Or designs d'enfant:

L. Schneps, Dessins d'enfants on the Riemann sphere. The Grothendieck theory of dessins d'enfants (Luminy, 1993), 47–77, London Math. Soc. Lecture Note Ser., 200, Cambridge Univ. Press, Cambridge, 1994.

I suppose that "non-compact complex algebraic curve" means complex affine curve.

The following counterexample was proposed by my friend Fedor Pakovich.

Let $D=\mathbf{C}\backslash\{-1,1\}$. Consider the $4$-th Chebyshev polynomial $$p_1(z)=2(2z^2-1)^2-1=8z^4-8z^2+1.$$ It has critical points at $0,\pm1/\sqrt{2}$, with critical values $\pm1$, therefore it defines an unramified covering $$p_1:C\to D,\quad\mbox{where}\quad C=\mathbf{C}\backslash p_1^{-1}(\{\pm1\}).$$ Now $p_2(z)=-p_1(z)$ is another unramified covering $C\to D$ of the same degree, but evidently $p_1\neq p_2\circ\phi.$

Remarks. 1. This example can be much generalized, of course; one can take any $D$ possessing a non-trivial automorphism $\psi$, then in most cases $p_1$ (mapping whatever Riemann surfce to $D$) and $p_2=\psi\circ p_1$ will not be related as stated in the problem.

The problem will become harder if under the same assumptions we relax the conclusion to $p_2=\psi\circ p_2\circ\phi.$ Fedor and I believe that counterexamples may still exist, but they will be rare.

I suppose that "non-compact complex algabraic curve" means complex affine curve.

Then the answer is negative. The simplest example is this: let $C=D=$ triply punctured sphere, and degree is $6$. (There is only one affine curve isomorphic to triply punctured sphere). There are two rational functions $p_1,p_2$ of degree $6$, each with three critical points and three critical values, such that the local degrees of $p_1$ at the critical points are $(3,5,5)$ and local degrees of $p_2$ are $(4,4,5)$. Evidently there is no isomorphism $\phi$ such that $p_1=p_2\circ\phi$.

Various constructions of such $p_1,p_2$ are available, see, for example A. Eremenko, A. Gabrielov, M. Shapiro and A. Vainshtein, Rational functions and real Schubert calculus, Proc. AMS, 134, 4 (2005) 949-957. Or one can use subgroups of the modular group. Or designs d'enfant.

I suppose that "non-compact complex algabraic curve" means complex affine curve.

Then the answer is negative. The simplest example is this: let $C=D=$ triply punctured sphere, and degree is $6$. (There is only one affine curve isomorphic to the triply punctured sphere). One can construct two rational functions $p_1,p_2$ of degree $6$, each with three critical points and three critical values, such that the local degrees of $p_1$ at the critical points are $(3,5,5)$ and local degrees of $p_2$ are $(4,4,5)$. Evidently there is no isomorphism $\phi$ such that $p_1=p_2\circ\phi$.

Various constructions of such $p_1,p_2$ are available, see, for example 

A. Eremenko, A. Gabrielov, M. Shapiro and A. Vainshtein, Rational functions and real Schubert calculus, Proc. AMS, 134, 4 (2005) 949-957.

Alternatively, one can use subgroups of the modular group. Or designs d'enfant:

L. Schneps, Dessins d'enfants on the Riemann sphere. The Grothendieck theory of dessins d'enfants (Luminy, 1993), 47–77, London Math. Soc. Lecture Note Ser., 200, Cambridge Univ. Press, Cambridge, 1994.