I know that the complex plane is a Lie group with +, but is it also a Lie group with the usual complex multiplication?
This would give us a nice geometrical interpretation of the famous Euler formula:
We have $exp:T_1\mathbb{C} \to \mathbb{C}$ and $T_1\mathbb{C} \cong \mathbb{C}$ because the latter is a linear space.
Also the complex multiplication is linear, so its differential is itself. Hence any left invariant vector field $X$ on $\mathbb{C}$ can be obtained by choosing a vector $X_0$ in $T_1\mathbb{C} \cong \mathbb{C}$ and using the formula
$$ X(z) = z X_0 $$
Now looking at the curve
$$ t \mapsto exp(tX_0) $$
in the case $X_0 = i$ give us the nice geometrical interpretation of the Euler Formula!