Start with the space $X_0=S^1$ (any connected manifold will do), and choose a basepoint in $X_0$. Construct a sequence of pointed spaces recursively as follows: $$X_{n+1}=\bigvee_{x,y\in X_n, x\ne y} X_n/
\Delta(x,y).$$ Let $X=\bigvee_{n=1}^\infty X_n$.

Then $X$ is connected and Hausdorff. Moreover, $X$ is naturally identified with a wedge sum with the following properties: 1. For each wedge summand, there are infinitely many other summands homeomorphic to it, 2. for any two points in a wedge summand $A$, there is a wedge summand homeomorphic to the quotient of $A$ by the two points, 3. For any two points belonging to different wedge summands $A, B$, there is a wedge summand homeomorphic to the quotient of $A\vee B$ identifying these points. From here it is easy to conclude that $X$ has the property that you want.