Two non-homeomorphic connected, hausdorff, locally compact spaces whose one-point compactifications are homeomorphic

Question

I'm looking for two non-homeomorphic connected, Hausdorff, locally compact spaces whose one-point compactifications are homeomorphic.

Without the connectedness property this is easy, for example: $[0,1) \cup (1,2]$ and $[0,2)$. I was thinking that I could maybe find two connected spaces which are locally compact but not compact where on loses the connectedness property when removing a point, and the other doesn't, but I can't seem to find such a space. Can anyone give an example of two such spaces?

Answer 1

Square without an interior point and square without a boundary point.

Answer 2

$$(\{(x,y):x^2+y^2=1\}\cup\{(x,0):-1\le x\le1\})\setminus\{(0,0)\}$$$$\text{and}$$$$(\{(x,y):x^2+y^2=1\}\cup\{(x,0):-1\le x\le1\})\setminus\{(1,0)\}.$$