>To have a notion that multiplication by imaginaries causes rotation, you'd need Euler's formula.

No you don't. You might notice this by simply working with imaginary numbers. You might invent imaginary numbers for rotation [0].

Alternatively, consider the multiplication (x + yi)(a + bi), as the value (a + bi) performing a transformation on (x + yi). We want (x + yi)(a + bi) = xy - by + ayi + bxi. If we consider (x + yi) to be a 2 dimensional matrix (with basis 1 and i), we can write the above equation as a matrix multiplication:

    [ x   y ] [  a   b ] = [ ax-by ; ay + bx]
                  -b  a

Notice that

    [  a   b  ] 
      -b  a

is just a rotation matrix multiplied by the scalar (a^2 + b^2).

[0] That is, define a group (in the group-theory sense) of functions of rotations, denoted as xi for real numbers x, and a group of group of functions for sliding, denoted x for real numbers x. You might then notice that you can combine these groups in a field structure, that happens to have 1i * 1i = -1, and that the subfield of elements with no i component happens to be isomorphic the the reals.