Any complex lattice admits a certain lattice basis

Question

A complex lattice consists of a pair of $\mathbb{R}$-linearly independent vectors of the real-vector space $\mathbb{C}$. We call two lattices $(\lambda_1,\lambda_2)$ and $(\mu_1,\mu_2)$ equivalent if $\mathbb{Z}\lambda_1+\mathbb{Z}\lambda_2= \mathbb{Z}\mu_1+\mathbb{Z}\mu_2$. Is any complex lattice equivalent to a lattice of the form $(\lambda_1,\lambda_2)$ with $\lambda_1\in \mathbb{R}$? Fo all the examples I have considered this holds but I am struggling to find a proof.

Answer 1

No: rotate the numbers $(1,i)$ in the complex plane by an angle with irrational tangent and you'll get $(\mu_1,\mu_2)$ such that the only real point of $\mathbb{Z}\mu_1+\mathbb{Z}\mu_2$ is the origin.