Separation axioms and when does there exist a connected space all whose proper connected subspaces are homeomorphic to the whole space?

Question

This question is in the spirit of this question Does every non-singleton connected metric space $X$ contains a connected subset (with more than one point) which is not homeomorphic with $X$? ; Fixing the convention that whenever we will talk about connected spaces or subsets we will talk about such with more than one point ; it is easy to observe that $\mathbb Z$ with co-finite topology fails to have the property mentioned in the question i.e. the space is connected and every proper connected subset is homeomorphic to the whole space . Now this counter example space is $T_1$ but not Hausdorff , nor regular , nor normal . So it is natural to ask : Does there exist a connected Hausdorff ( resp. for Regular , $T_3$ , Normal , $T_4$ ) space such that every proper connected subspace of it is homeomorphic to the whole space ?