let's start with a definition of i,

The square root of -1 is a special number we call i and it is a member of a set of numbers called the imaginary numbers.  The imaginary numbers are the set of all numbers that are the square root of negative numbers.
- The Math Forum

Q: why bother with such a seemingly frivolous and useless abstraction as i?
A: albeit abstract, i is neither frivolous nor useless, as is demonstrated below,

For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as 2/3 and 1/8  are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones.

Negative numbers such as -3 and -5 are meaningless when weighing the mass of an object, but essential when keeping track of monetary debits and credits.

Similarly, imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, cartography, and many others.
-wikipedia

yet despite their utility, it is the highly abstract nature of
imaginary numbers that we tend to struggle with, to resist

for most of us this level of abstraction represents a barrier
that separates us from the world of higher mathematics

while for others, as shown in the video below, the abstraction
of imaginary numbers is "cool"

and it represents the first real and welcome challenge on the
road to a career in mathematics, and more importantly, to
a glimpse at truths that few can comprehend



the soundtrack of the above is an excerpt from a
BBC Radio 4 5 Numbers podcast

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- The World’s Most Mysterious Number
- The Fibonacci Sequence and Nature’s Number
- The Most Important Number In the Universe
- A Hierarchy of Infinities
- Are Numbers Discovered Or Invented?