Think of the smallest number greater than 0...

[o_O]

Without them, electrical engineering wouldn't exist - taking complex mathematics away from that is like taking half of a chef's ingredients away.

Yeah, if you take the squizzleberries and xarkon beans away from the chef, he surely can't make his imaginary dish anymore.

That doesn't make any sense. A chef's profession is not based or dependent upon dishes made from squizzleberries and xarkon beans or any other imaginary ingredient. Chefs don't cook imaginary dishes. If you take the imaginary stuff away, a chef can still be a chef without any impact on the quality of their work.

By the same token, electrical engineering is heavily reliant upon complex numbers. You take them away, and an electrical engineer can no longer electrically engineer.

You can teach computers to use concepts that don't exist as long as these concepts have a solid set of rules to their use. My brother's original major was in engineering, so he knows how "important" imaginary numbers are. Obviously not important enough for his classes to learn them. Even if his students do plan becoming electrical engineers.

This, I just downright don't believe.

Firstly, no, you cannot build a computer without mathematics. The theory and logic that goes into electrical physics is very complex and you can't just <i>build</i> something that works. It's not about teaching computers to do stuff, it's about designing a set of circuits that perform a task - in this case an extremely complex task such as computing.
Secondly, not all branches of mathematics involve complex maths. It is not possible to complete even one year of an electrical engineering degree without using complex maths enough to know that it is essential. It just isn't possible. If he doesn't teach them, I can only assume that he's either not teaching calculus, or that the calculus he is teaching isn't advanced enough to include complex numbers. Note that I'm not disputing the fact that he was an engineering major.