Newman (1963) proved the following.
Theorem 1. Let $d \in \mathbb{N}$. Define $$p(x) = \prod_{k=0}^{d-1} \left(x+\exp\left(\frac{-k}{\sqrt{d}}\right)\right)$$ and $$r(x) = \frac{\sqrt{x} \cdot (p(\sqrt{x}) - p(-\sqrt{x}))}{p(\sqrt{x}) + p(-\sqrt{x})}.$$ Then $r(x)$ is a rational function of degree $\lceil d/2 \rceil$ with real coefficients. (Specifically, $r(x)$ is the ratio where the numerator is a polynomial in $x$ of degree $\lceil d/2 \rceil$ and the denominator is a polynomial in $x$ of degree $\lfloor d/2 \rfloor$.) Furthermore, $$\sup_{x \in [0,1]} |r(x)-\sqrt{x}| \le 3 \cdot \exp(-\sqrt{d}).$$
Subsequent work has obtained optimal constants and generalized to powers other than the square root; see Stahl (1993) and references therein.
I'm interested in extending to the complex plane. I.e., rather than $x \in [0,1]$, I want the approximation guarantee to hold for all $x \in \mathbb{C}$ with $|x-\frac12|\le\frac12$. The following gives a specific strong conjecture.
Conjecture 2. There exists a universal constant $c>0$ such that the following holds. Let $d \in \mathbb{N}$. There exists a rational function $r(x)$ of degree $\le d$ with real coefficients such that $$\sup \left\{ \left| r(x) - \sqrt{x} \right| : x \in \mathbb{C}, \left|x-\frac12\right|\le\frac12 \right\} \le \exp(-c \cdot \sqrt{d} +1/c).$$
(Some technicalities: Implicit in Conjecture 2's conclusion is that $r(x)$ has no poles with $|x-\frac12|\le\frac12$. We take the principal branch of the square root so that it matches Newman's result on the positive real interval $[0,1]$ and the square root is continuous in the region of interest, since the branch cut is on the negative real axis.)
Numerically, it seems that Newman's construction works on the complex plane without modification. Here is a plot of the error as we go around the boundary of the disc for varying degree.
Here is a plot showing that the error is maximal at the boundary. That is, we vary the radius of the circle centered at $\frac12$ and compute the maximum error on that circle.
Finally, here is a plot showing how the error decreases with degree. This seems consistent with $\exp(-c\sqrt{d})$ asymptotic error.
For my application, it would suffice to prove something weaker than Conjecture 2 above. In particular, we can shrink the disc slightly, as in Conjecture 3 below. It would also suffice to prove some kind of weighted average error bound. But the statement becomes messy, so I'll leave it at this.
Conjecture 3. There exists a universal constant $c>0$ such that the following holds. Let $d \in \mathbb{N}$. There exists a rational function $r(x)$ of degree $\le d$ with real coeficients such that $$\sup \left\{ \left| r(x) - \sqrt{x} \right| : x \in \mathbb{C}, \left|x-\frac12\right|\le\frac12-\varepsilon \right\} \le \varepsilon,$$ where $\varepsilon = \exp(-c \cdot \sqrt{d} +1/c)$.
This seems like a question that should have been studied. Any pointers or suggestions would be greatly appreciated.