  In this paper, we discuss complete minimal immersions in
$\mathbb{R}^N$($N\geq4$) with finite total curvature and embedded planar ends.
First, we prove nonexistence for the following cases: (1) genus 1 with 2
embedded planar ends, (2) genus $\neq4$, hyperelliptic with 2 embedded planar
ends like the Lagrangian catenoid. Then we show the existence of embedded
minimal spheres in $\mathbb{R}^4$ with 3 embedded planar ends. Moreover, we
construct genus $g$ examples in $\mathbb{R}^4$ with $d$ embedded planar ends
such that $g\geq 1$ and $g+2\leq d\leq 2g+1$. These examples include a family
of embedded minimal tori with 3 embedded planar ends.
