We start with the stochastic differential equation $$\mathrm{d}X_t=\lambda X_t\mathrm{d}t + \sigma X_t \mathrm{d}W_t,$$ where $\lambda,\sigma\in\mathbb{C}$.
I would like to apply Itô formula to obtain an equation for $\mathrm{d}\left|X_t\right|^2.$
Here's my attempt (using the fact that $|X_t|^2 = \mbox{Re}(X_t)^2+\mbox{Im}(X_t)^2$): $$\mathrm{d}\left|X_t\right|^2=2\lambda X_t \Big(\mbox{Re}(X_t)+\mbox{Im}(X_t)\Big)\mathrm{d}t + |\sigma|^2\left|X_t\right|^2\mathrm{d}t+\mathrm{d}M_t,$$ where $\mathrm{d}M_t$ is the martingale term.
However, the notes I'm looking at say that we get $$\mathrm{d}\left|X_t\right|^2=(2\mbox{Re}(\lambda)+|\sigma|^2)\left |X_t\right|^2\mathrm{d}t+\mathrm{d}M_t.$$
How did they get that, and how does the real part of $\lambda$ come into play?
If we treat $X$ as an $\mathbb{R}^2$-valued process, Itô's lemma says the relevant drift term is given by $\nabla f(X_t) \cdot \lambda X_t dt$.
You're right that $\renewcommand{\Re}{\operatorname{Re}}\renewcommand{\Im}{\operatorname{Im}}\nabla f(X_t) = (2 \Re(X_t), 2 \Im(X_t))$. But then $$\lambda X_t = (\Re(\lambda) \Re(X_t) - \Im(\lambda) \Im(X_t), \Re(\lambda) \Im(X_t) + \Im(\lambda) \Re(X_t)).$$ When you dot these, you find that the $\Im(\lambda)$ terms cancel and leave you with $$2 \Re(\lambda)(\Re(X_t)^2 + \Im(X_t)^2) = 2 \Re(\lambda) |X_t|^2.$$