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Does every proof in the complex numbers also prove the statement in the real numbers? I thought it might be true, because the real numbers are part of the complex numbers.

If we show something is true for every complex number $zin mathbb{C}$, then we have shown that it is true for every $xin mathbb{R}$ since $mathbb{R}subsetmathbb{C}$.

However, not "every proof in the complex numbers" is of this form. For instance, consider the following example. We can show that there exists a $zin mathbb{C}$ such that $z^2=-1$, but there is no real number that has this property.

Your reasoning that if something is true about complex numbers it must be true about reals because reals are complex is sound. But I don't think you are thinking through just what sort of statements can be proven.

This is not a complete answer but you need to think about "some" and "all".

If X is true for all complex, it is true for all reals.

If Y is true for some complex, it may or may not be true for some, all, or no reals.

If W is true for no complex, it is not true for any reals.

If A is true for all reals then it is true for some complex. It may or may not be true for all conplex.

If B is true for some reals then it is true for some complex.

And if C is not true for any real it might or might not be true for some complex (but definitely not true for all).

There is a problem in Kreyszig's Functional Analysis:

Let $X$ be an inner product space over $mathbb{C}$, and $T:Xto X$ is a linear map. If $langle x,Txrangle =0:forall xin X$, then $T$ is the null transformation.

This is not true for inner product spaces over $mathbb R$.

This depends on what you mean by a proof in the reals. Not every statement about the complex numbers is a statement about the reals. However, you can think of every complex number as a pair of real numbers (a,b), with a corresponding relationship, where one defines complex addition and multiplication as operations on ordered pairs of real numbers. So any statement you make about the complex numbers does correspond to a statement about real numbers with twice as many variables.

Other answers have already said that the validity of this statement depends on what you mean by a "proof". Here's another example of when something is true in $mathbb C$ but not in $mathbb R$: the fundamental theorem of algebra. It states that a polynomial of degree $n$ always has exactly $n$ roots in $mathbb C$, which is however clearly not true in $mathbb R$ except degenerate cases. In fact, if FTA were to be true in $mathbb R$, it would be a strictly stronger statement and would imply FTA in $mathbb C$, not the other way round.

It is more or less dangerous to think of proofs in complex analysis in real sense, because things in complex analysis are vastly different from those in real analysis.

A function mapping from $mathbb{C}$ to $mathbb{C}$ that we concern in complex analysis is usually in terms of $z in mathbb{C}$ instead of individual $x, y in mathbb{R}$ even though we have $z = x+yi$ as usual. A differentiable (holomorphic) function in complex sense is a much stronger. Some results could be surprising for those who just started learning complex analysis.

For example, a real differentiable function may not be twice differentiable, and its derivative may not even be continuous (that's why we have different regularity conditions like $C^1$, $C^2$ up to $C^infty$ and $C^omega$). However, holomorphic functions mapping from $mathbb{C}$ to $mathbb{C}$ are automatically differentiable infinitely many times. Liouville's Theorem states that any bounded entire (holomorphic in $mathbb{C}$) function is constant. This is certainly not true in real analysis: how would real analysis become if all bounded functions mapping from $mathbb{R}$ to $mathbb{R}$ that are differentiable on $mathbb{R}$ (e.g. $sin$, $cos$, $arctan$) are constant?