Lagrangian multipliers in complex optimization

This topic has probably long been forgotten, but in the interest of the potential future readers let me just point out the fact that the OP's approach did not fail at all. It seemed to fail for the trivial reason that the derivatives were calculated incorrectly. Using the OP's notation (and the convention of regarding $z$ and $z^*$ as independent variables), we indeed have $$\partial\mathcal{L}/\partial z = z^* + \lambda = 0,$$ $$\partial\mathcal{L}/\partial z^*= z - \lambda = 0.$$ Adding both equations together we find $z+z^* = 0$, and substituting this result back into $z-z^*-\mathrm{i} = 0$, we obtain $2z - \mathrm{i} = 0$, the desired answer.

Let me also note that the expression $\lambda(z - z^* - \mathrm{i})$ is real because it is equal to zero (which is a real number) in the whole domain of the complex plane that we are constrained to by the very definition of the problem.

I'm afraid there are myriad problems with your approach. You're not taking derivatives with respect to complex $z$ correctly; indeed, the fact that you took two, with respect to both $z$ and $z^*$, is suspect---those are the same variable! And you have to be careful building the Lagrangian as well.

The simplest way to do this is to write the equivalent real-valued problem: \begin{array}{ll} \text{minimize} & z_R^2 + z_I^2 \\ \text{subject to} & z_R = z_R + 0 \\ & z_I = - z_I + 1 \\ \end{array} Of course, at this point, we can see the answer by inspection: $z_R=0$, $z_I=1/2$. But if we're crazy enough to build the Lagrangian, we can do this: $$L(z_R,z_I,\lambda_R,\lambda_I) = z_R^2+z_I^2-\lambda_R\cdot 0 - \lambda_I\cdot(2z_I-1)$$ Then the optimality conditions are $$2z_R = 0 \qquad 2z_I - 2\lambda_I = 0 \qquad 2z_I - 1 = 0$$ and again we get $z_R=0$, $z_I=1/2$. $\lambda_R$ is arbitrary, and $\lambda_i = z_I=1/2$.

Now let's show how to do this as in the complex domain. The Lagrange multiplier $\lambda$ is complex, because the equality constraint is complex. But the inner product we need to use is a real inner product $\langle a,b\rangle=\Re(a^*b)$, because the Lagrangian is a real expression: $$L(z,\lambda) = \langle z, z \rangle - \langle \lambda, z-z^*-1 \rangle = z^* z - \Re(\lambda^*(z-z^*-1))$$ So how the heck do you differentiate with respect to complex $z$? Well, I wish I could tell you a good resource here. But what I do is treat it as a multivariate function (because in a real sense, it is), and look at it as a gradient. For linear and quadratic forms, we have $$\nabla_z \langle a, z \rangle = \nabla_z \Re(a^*z) = \nabla_z \Re(az^*) = a, \quad \nabla_z (z^*z) = 2z.$$ So the optimality condition is $$2z - \lambda + \lambda^* = 2z - 2i\Im(\lambda) = 0 \qquad 2z = i$$ and we have $z=\tfrac{1}{2}i$ and $\Im(\lambda)=1/2$, with $\Re(\lambda)$ arbitrary.

I'll be honest, it will not surprise me if someone comes along and criticizes my approach to the calculus in the complex version. I don't think I've ever been given formal instruction, I just derived what is necessary to get the optimization correct. The key here is that it must be mathematically equivalent to the real version if it's going to make any sense. Optimization only occurs with real objectives and real Lagrangians, so the multipliers and inner products and derivatives must be defined to get you there.