A high school student asked me whether learning Binomial theorem would help him with complex numbers. At first, I said no but then I realised I was too rusty in the subject to make a judgement.
What should I tell him? If there is some relevance of Binomial Theorem in complex numbers, to what extent?
Also, specific examples where Binomial Theorem is relevant to complex analysis would be great!
You say "must study" but I don't think it's the right way to look at it. Think of mathematical subjects as tools in a toolkit. As you progress, you gradually accumulate these tools. You never know when a particular tool may come in handy. Before starting on complex number theory (not even analysis), I would say being experienced with all the tools pertaining to basic algebra is important. I think most people would consider binomial expansions (integer powers) to be part of that basic algebra toolkit. So a resounding "yes" from me.
Here are a couple of examples where your student may need to apply binomial theorem when dealing with complex numbers at even an elementary level.
Example 1:
Simplify $(2+3i)^4$, expressing the result in the form $a + bi$.
There is more than one way to handle this problem, and a slightly sophisticated approach would be to transform the number into exponential form before taking the power. However, for such a low exponent, a simple application of binomial theorem is probably easiest. $(2 + 3i) ^4 =2^4 + \binom 4 12^3(3i) + \binom 4 22^2(3i)^2 + \binom 4 32(3i)^3 + (3i)^4$ which can then be easily simplified by evaluating and grouping the real and complex terms together.
Example 2:
As your student delves deeper, he/she will cover de Moivre's theorem, which affords a very easy way to take powers of complex numbers. It's especially useful in the computation of the multiple angle formulae in trigonometry.
For example, suppose you wanted to compute $\cos 3\theta$ in terms of powers of $\cos\theta$.
de Moivre's theorem allows you to immediately state that $(\cos\theta + i\sin\theta)^3 = \cos 3\theta + i\sin 3\theta$
Since the real parts on both sides have to be equal, you can use binomial expansion to determine and simplify the real part of the LHS, and then derive $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ without much fuss.
The real power comes when you want to generalise this to any multiple $n$, as in this Wolfram article on Multiple Angle formulae