My question is similar to this one but hopefully simpler. See the attached image below I created on Desmos. 
Now, the zeroes for the blue parabola are $x = -1$ and $x = 1$. The zeroes for the red parabola are $+i$ and $-i$. The geometric interpretation for the former is easily understood as the intersection of the parabola with the x axis. But I am struggling to find a geometric interpretation for the latter. Based on Angae MT's comments, I took another stab at drawing this, where the z axis is imaginary and the BLUE parabola represents the reflection that Angae is speaking of and the GREEN represents the same parabola but rotated 90 degrees. The only difference is that it is plotted in 3D with the 3rd axis being imaginary. I tried to upvote MT Angae's answer but I don't have sufficient reputation.
3D of Angae's Answer
I give a proof over open upward parabola, which can be generalised to open downward parabola too (treat it as exercise!)
Lemma. Let $f(x)=(x+h)^2+k$ be an open upward parabola such that $f(x)=0$ has roots $a\pm bi$, where $a,b\in\mathbb{R}$. Define $g(x):=k-(x+h)^2$, then $g(x)=0$ has roots $a\pm b$.
Proof. By quadratic formula, $$g(x)=0\iff x=\dfrac{2h\pm\sqrt{(-2h)^2-4(-1)(k-h^2)}}{-2}=-h\pm\sqrt{4k}$$ But by $f(x)$, we know $a+bi=\dfrac{-2h\pm\sqrt{(2h)^2-4(h^2-k)}}{2}=-h\pm\sqrt{-4k}$. So we have $$\begin{cases}a=-h\\ b=\sqrt{4k}\end{cases}$$ And it follows that $g(x)=0$ has roots $a\pm b$.
What is the use of this Lemma? The idea is the graph of $g(x)$ is actually obtained by reflecting the graph of $f(x)$ along the horizontal line which touches the vertex of $y=f(x)$. This gives the geometric meaning.
For example, for your $x^2+1$, reflecting the curve $y=x^2+1$ along $y=1$, then it would intersects $x$-axis at $(-1,0)$ and $(1,0)$ so the original function has zeroes $0+1i$ and $0-1i$.
For another example like $x^2+6x+100$, reflecting the curve along $y=100-3^2=91$, you get $91-(x+3)^2$. Then it has the root $$-3\pm\sqrt{91}$$ So $x^2+6x+100=0$ gives the root $-3\pm\sqrt{91}i$.
Hope this answer is useful to you :)