Information/references/examples on fields $\mathbb R^3 \to \mathbb C^3$ with divergence and curl free real and imaginary parts

Question

In the course of some physical considerations I came across a complex vector field $$ \mathbf u = \mathbf v + i \mathbf w, $$ with \begin{align} \mathbf v:& \mathbb R^3\to \mathbb R^3\\ \mathbf w:& \mathbb R^3\to \mathbb R^3 \end{align}

and the special propert, that it has a divergence-free imaginary and curl free real components, that means

\begin{align} \vec{\nabla}\times \mathbf v & = \mathbf 0 \tag{1}\\ \vec{\nabla}\cdot\mathbf w & = 0 \tag{2} \end{align}

In my attempts to better understand and interpret the quantitiy described by this field, I started to wonder if:

Question 1: This property has a special name/term in complex vector analysis.

Question 2: If there are any important prominent examples of such fields.

Question 3: If there are other properties that follow as a consequence of (1) and (2).

Question 4: If anyone can hint me at literature where such fields are investigated.

I would be very grateful for any hints at this. I would hope for answers like those to this question.