There are two definitions of intersection multiplicity of two complex algebraic curves. One is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t^k) )$ be the local normalization then $(V_g \cdot V_f)_P = ord_P(g(t^k, y(t^k)))$. And one is given in Algebraic geometry by Hartshorne. $(V_g \cdot V_f)_P = \mathrm{length}(O_p/(f,g))$ as $O_P$ module.

Are these definitions coincide?

There may be some approaches. Showing that completion preserves length may help cause you can thereby catch the normalization in complex analysis to algebraic one. Showing that $\mathrm{length}(O_P/(g, f_1 f_2))= \mathrm{length}(O_P/(g, f_1)) \mathrm{length}(O_P/(g, f_1))$ is also important.

There are two definitions of intersection multiplicity of two complex algebraic plane curves. One of them is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t) )$ be the local normalization of $V_f$ then $(V_g \cdot V_f)_P = ord_P(g(t^k, y(t)))$. The other one is given in Algebraic Geometry by Hartshorne. $(V_g \cdot V_f)_P = \mathrm{length}(O_p/(f,g))$ as $O_P$ module.

Do these definitions coincide?

There may be some approaches. Showing that completion preserves length may help cause you can thereby catch the normalization in complex analysis to algebraic one. Showing that $\mathrm{length}(O_P/(g, f_1 f_2))= \mathrm{length}(O_P/(g, f_1))+\mathrm{length}(O_P/(g, f_2))$ is also important.

There are two definitions of intersection multiplicity of two complex algebraic curves. One is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t^k) )$ be the local normalization then $(V_g \cdot V_f)_P = ord_P(g(t^k, y(t^k)))$. And one is given in Algebraic geometry by Hartshorne. $(V_g \cdot V_f)_P = \mathrm{length}(O_p/(f,g))$ as $O_P$ module.

Are these definitions coincide?

There are two definitions of intersection multiplicity of two complex algebraic curves. One is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t^k) )$ be the local normalization then $(V_g \cdot V_f)_P = ord_P(g(t^k, y(t^k)))$. And one is given in Algebraic geometry by Hartshorne. $(V_g \cdot V_f)_P = \mathrm{length}(O_p/(f,g))$ as $O_P$ module.

Are these definitions coincide?

There may be some approaches. Showing that completion preserves length may help cause you can thereby catch the normalization in complex analysis to algebraic one. Showing that $\mathrm{length}(O_P/(g, f_1 f_2))= \mathrm{length}(O_P/(g, f_1)) \mathrm{length}(O_P/(g, f_1))$ is also important.