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Here's a closer shot of the plant. My wife says it is Bleeding Heart but probably of a different variety. But the leaves do look like a begonia plant but the flowers are much smaller and different shaped as begonia.
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Definitely a Begonia :) There are different types. Some have nice, big flowers that look almost like roses. Some have like yours.
They have beautiful leaves.

I like the one at the back too. Is that Bougainvillea?

Yes you are correct. My wife is going to be heartbroken about this :(

The plant shop where she bought this a long time ago told her it was Bleeding Hearts. But the dead giveaway are the leaves which are that of Begonias.

Yes that is a bougainvillea at the background. Here's a closer look.
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Begonias are also very pretty plants and can grow quite big!

That is lovely! I see them growing around here, but I dont have space to have one.