is there a bijection for $f: \mathbb R \to \mathbb C$?

Question

I imagine no since the dimensions do not match

but they have the same cardinality $|\mathbb R |= |\mathbb C|$?

Answer 1

If you just want a bijection as sets, the answer is yes. That’s exactly what it means for two sets to have the same cardinality. If you’re looking for a bijection that preserves some structure, that will depend on precisely what you’re trying to preserve but the answer is very probably no.