3 corrected 'embedded' to 'submanifold'

Your question already has the answer in it for $n=2$. Take a connected complex curve $L\subset\mathbb{C}^2$ that is dense in $\mathbb{C}^2$. Then $L$ is Lagrangian for the real part of the holomorphic $2$-form $\Upsilon = dz^1\wedge dz^2$. This real part of $\Upsilon$ is equivalent to the standard symplectic structure on $\mathbb{R}^4$ by a linear change of variables.

Added comment about embeddingsinjectivity: Note, by the way, that one can easily arrange for such an $L$ to be embeddeda submanifold, not just immersedthe image of an immersion (i.e., the immersion is injective). One simple explicit way to do this is to select constants $\lambda_1,\ldots,\lambda_k$ such that the subgroup in $\mathbb{C}^\times$ generated by the numbers $\mathrm{e}^{2\pi i\lambda_1},\ldots,\mathrm{e}^{2\pi i\lambda_k}$ is dense in $\mathbb{C}^\times$ and consider the linear differential equation $$\frac{dy}{dx} = \left(\frac{\lambda_1}{x-x_1}+\cdots + \frac{\lambda_k}{x-x_k}\right)\ y$$ where $x_1,\ldots,x_k\in \mathbb{C}$ are distinct. The graph of any nonzero multi-valued solution $y(x)$ over $\mathbb{C}\setminus\{x_1,\ldots,x_k\}$ will then be dense in $\mathbb{C}^2$. (Consider the holonomy around the punctures $x_j$.) Of course, these graphs are the Riemann surfaces associated to the multivalued functions $$y = y_0 (x{-}x_1)^{\lambda_1}\cdots(x{-}x_k)^{\lambda_k}$$ (when $y_0\not=0$). These are obviously integral curves (leaves) of the polynomial $1$-form $$\omega = (x{-}x_1)\cdots(x{-}x_k)\ dy - q(x) y\ dx$$ for some polynomial $q$ of degree at most $k{-}1$ in $x$. Aside from the obvious closed leaves $x-x_j=0$ and $y=0$, the rest of the leaves are embedded and dense submanifolds. (This just gives a simple, explicit example of the general theorem that Richard quoted.)

Dense analytic curves in $\mathbb{R}^2$: It is not hard to construct dense, connected analytic curves in $\mathbb{R}^2$: There exist analytic metrics on the $2$-sphere that have geodesics that wander densely over the surface. Now take such a geodesic and remove a point from $S^2$ through which the geodesic doesn't pass. What's left is a dense analytic curve in $\mathbb{R}^2$. If you are willing to use Finsler metrics, you can even do this with a rotationally invariant real analytic Finsler metric on the $2$-sphere (eg. Katok's examples), so that you can write down the dense analytic curve very explicitly.

I'll think about the case $n>2$. I don't see it yet either, but maybe it's not too hard.

Added comment about embeddings: Note, by the way, that one can easily arrange for such an $L$ to be embedded, not just immersed. One simple explicit way to do this is to select constants $\lambda_1,\ldots,\lambda_k$ such that the subgroup in $\mathbb{C}^\times$ generated by the numbers $\mathrm{e}^{2\pi i\lambda_1},\ldots,\mathrm{e}^{2\pi i\lambda_k}$ is dense in $\mathbb{C}^\times$ and consider the linear differential equation\frac{dy}{dx} = \left(\frac{\lambda_1}{x-x_1}+\cdots + \frac{\lambda_k}{x-x_k}\right)\ ywhere $x_1,\ldots,x_k\in \mathbb{C}$ are distinct. The graph of any nonzero multi-valued solution $y(x)$ over $\mathbb{C}\setminus\{x_1,\ldots,x_k\}$ will then be dense in $\mathbb{C}^2$. (Consider the holonomy around the punctures $x_j$.) Of course, these are the Riemann surfaces associated to the multivalued functionsy = y_0 (x{-}x_1)^{\lambda_1}\cdots(x{-}x_k)^{\lambda_k}(when $y_0\not=0$). These are obviously integral curves (leaves) of the polynomial $1$-form\omega = (x{-}x_1)\cdots(x{-}x_k)\ dy - q(x) y\ dxfor some polynomial $q$ of degree at most $k{-}1$ in $x$. Aside from the obvious closed leaves $x-x_j=0$ and $y=0$, the rest of the leaves are embedded and dense. (This just gives a simple, explicit example of the general theorem that Richard quoted.)
Dense analytic curves in $\mathbb{R}^2$: It is not hard to construct dense connected analytic curves in $\mathbb{R}^2$: There exist analytic metrics on the $2$-sphere that have geodesics that wander densely over the surface. Now take such a geodesic and remove a point from $S^2$ through which the geodesic doesn't pass. What's left is a dense analytic curve in $\mathbb{R}^2$. If you are willing to use Finsler metrics, you can even do this with a rotationally invariant real analytic Finsler metric on the $2$-sphere (eg. Katok's examples), so that you can write down the dense analytic curve very explicitly.
Your question already has the answer in it for $n=2$. Take a connected complex curve $L\subset\mathbb{C}^2$ that is dense in $\mathbb{C}^2$. Then $L$ is Lagrangian for the real part of the holomorphic $2$-form $\Upsilon = dz^1\wedge dz^2$. This real part of $\Upsilon$ is equivalent to the standard symplectic structure on $\mathbb{R}^4$ by a linear change of variables.
I'll think about the case $n>2$. I don't see it yet either, but maybe it's not too hard.