= Solution
A <variational symmetry of a Lagrangian density> maps admissible field configurations to configurations with the same <action>, up to an allowed boundary contribution. Thus (i) the <action> is invariant for suitable <boundary conditions>, while (ii) the <Lagrangian density> need only change by a <total divergence>, rather than be pointwise invariant. For an <internal symmetry of a classical field theory>, the coordinates are fixed. A spacetime transformation also changes the integration measure and must be treated accordingly.
A <conserved current> satisfies $\partial_\mu j^\mu=0$ <on shell>. The <Noether charge> $Q=\int j^0\,d^3x$ then obeys $\dot Q=-\int_{\partial V}\mathbf j\cdot d\mathbf S$, so conservation of the integrated charge additionally requires vanishing net boundary flux.
For several real field components $q_A$ and a constant infinitesimal parameter, write $\delta q_A=\varepsilon X_A$ and $\delta\mathcal L=\varepsilon\partial_\mu K^\mu$. The <product rule> gives
$$
\frac{\delta\mathcal L}{\varepsilon}=\sum_A\left(\frac{\partial\mathcal L}{\partial q_A}-\partial_\mu\frac{\partial\mathcal L}{\partial(\partial_\mu q_A)}\right)X_A+\partial_\mu\left(\sum_A\frac{\partial\mathcal L}{\partial(\partial_\mu q_A)}X_A\right).
$$
Subtract the specified divergence. The <Euler-Lagrange field equations> make the first term vanish, proving the <Noether theorem> with
$$
\boxed{j^\mu=\sum_A\frac{\partial\mathcal L}{\partial(\partial_\mu q_A)}X_A-K^\mu,\qquad \partial_\mu j^\mu=0.}
$$
For a <complex scalar field>, treat the field and its <complex conjugate> as independent variables in this calculation; this is equivalent to using its two real components. This is the <Noether current for a first-derivative scalar field>, summed over components.
Collect the two complex fields into $\Phi=(\phi_1,\phi_2)^T$. Their <Lagrangian density> has the radial form
$$
\mathcal L=\partial_\mu\Phi^\dagger\partial^\mu\Phi-m^2\Phi^\dagger\Phi-\lambda(\Phi^\dagger\Phi)^2.
$$
For a constant <Hermitian matrix> $T$, the <unitary transformation> $\delta\Phi=i\varepsilon T\Phi$ preserves both the kinetic contraction and $\Phi^\dagger\Phi$. In particular the two specified transformations have generators $T_1=\sigma_1$ and $T_2=\sigma_2$, the first two <Pauli matrices>. Their <Noether currents> are instances of the <unitary scalar multiplet current>
$$
j_T^\mu=i\left[(\partial^\mu\Phi^\dagger)T\Phi-\Phi^\dagger T\partial^\mu\Phi\right].
$$
Explicitly,
$$
\begin{aligned}
j_1^\mu&=i\left(\phi_2\partial^\mu\phi_1^*+\phi_1\partial^\mu\phi_2^*-\phi_2^*\partial^\mu\phi_1-\phi_1^*\partial^\mu\phi_2\right),\\
j_2^\mu&=\phi_2\partial^\mu\phi_1^*-\phi_1\partial^\mu\phi_2^*+\phi_2^*\partial^\mu\phi_1-\phi_1^*\partial^\mu\phi_2.
\end{aligned}
$$
An overall change of current sign corresponds to reversing the associated symmetry parameter; the signs here follow the two transformations as written.
The <matrix commutator> of their infinitesimal matrices is $[i\sigma_1,i\sigma_2]=-2i\sigma_3$. The antisymmetric difference of successive transformations therefore supplies a third symmetry, after a nonzero rescaling of its parameter:
$$
\delta\phi_1=i\zeta\phi_1,\qquad\delta\phi_2=-i\zeta\phi_2.
$$
Its current is
$$
\boxed{j_3^\mu=i\left(\phi_1\partial^\mu\phi_1^*-\phi_1^*\partial^\mu\phi_1-\phi_2\partial^\mu\phi_2^*+\phi_2^*\partial^\mu\phi_2\right).}
$$
The three <traceless matrices> $\sigma_a/2$ obey $[\sigma_a/2,\sigma_b/2]=i\epsilon_{abc}\sigma_c/2$. Their exponentials act as <SU(2)> on the complex doublet. This identifies both the <Lie algebra> and the group acting on these fields: the central element $-I$ changes $\Phi$ and is not the identity transformation, so the doublet action is not an action of $SO(3)$ obtained by identifying $I$ with $-I$.
There is also a common <global phase symmetry of a complex scalar field>, $\Phi\mapsto e^{i\alpha}\Phi$, with current $j_I^\mu$. Together with <SU(2)> this gives $U(2)\simeq(SU(2)\times U(1))/\mathbb Z_2$ among complex-linear transformations. Independent constant phases of $\phi_1$ and $\phi_2$, and field interchange, are already contained in this group.
In fact this particular <scalar potential> has a larger real internal symmetry. Write $\phi_1=(q_1+iq_2)/\sqrt2$ and $\phi_2=(q_3+iq_4)/\sqrt2$; then
$$
\mathcal L=\frac12\sum_{A=1}^4\partial_\mu q_A\partial^\mu q_A-\frac{m^2}{2}\sum_Aq_A^2-\frac\lambda4\left(\sum_Aq_A^2\right)^2.
$$
The <O(4) symmetry of a complex scalar doublet> includes transformations mixing $\Phi$ with $\Phi^*$, which need not be complex-linear. Its connected part has six rotation generators, so listing $U(2)$ alone would not exhaust the internal symmetries of the displayed isolated theory. Individual field conjugations are examples of its discrete transformations. Finally the spacetime-independent scalar coefficients and Lorentz contractions give <Poincaré symmetry>, and the real scalar action also permits spatial reflection and time reversal. These statements concern the action; a chosen vacuum or external <boundary conditions> can preserve fewer symmetries. Special parameter choices can introduce additional symmetries, but are not required here.
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