Nope. (Just to imitate your style)

There's more to it than rotation by 180 degrees. More pedagogically ...

Define a tuple (a,b) and define addition as pointwise addition. (a, b) + (c, d) = (a+c, b+d). Apples to apples, oranges to oranges. Fair enough.

How shall I define multiplication, so that multiplication so defined is a group by itself and interacts with the addition defined earlier in a distributive way. Just the way addition and multiplication behave for reals.

Ah! I have to define it this way. OK that's interesting.

But wait, then the algebra works out as if (0, 1) * (0, 1) = (-1, 0) but right hand side is isomorphic to -1. The (x, 0)s behave with each other just the way the real numbers behave with each other.

All this writing of tuples is cumbersome, so let me write (0,1) as i.

Addition looks like the all too familiar vector addition. What does this multiplication look like? Let me plot in the coordinate axes.

Ah! It's just scaled rotation, These numbers are just the 2x2 scaled rotation matrices that are parameterized not by 4 real numbers but just by two. One controls degree of rotation the other the amount of scaling.

If I multiply two such matrices together I get back a scaled rotation matrix. OK, understandable and expected, rotation composed is a rotation after all. But if I add two of them I get back another scaled rotation matrix, wow neato!

Because there are really only two independent parameters one isomorphic to the reals, let's call the other one "imaginary" and the tupled one "complex".

What if I negate the i in a tuple? Oh! it's reflection along the x axis. I got translation, rotation and reflection using these tuples.

What more can I do? I can surely do polynomials because I can add and multiply. Can I do calculus by falling back to Taylor expansions ? Hmm let me define a metric and see ...