Hi Jouko,
I echo some of the comments on presentation and also mathematical application.
i is a placeholder for a number. However it represents a specific dimensionless value, not c or h.
X^2 1^2 = i^2 * 0^2 = 0 and then x^2 = (-1)^2, which has the solution x = sqrt(-1) = i. In this situation, x is not a variable, but an unknown to be solved.
To consider a^2 b^2 = c^2 as an area, the values of the variables have to be allowed to vary. This is not the case for the solution of x^2 1 = 0 (where x is an unknown, not a variable).
In a different direction, since 'i' is really just a placeholder for an unknown, there is nothing to prevent us evaluating that unknown and defining an actual numeric representation of both i and, by extension, any complex number. In other words, i should be able to be represented in the same way as any other (real) number is - as a single value, rather than x iy. This would require some new areas of mathematics (like maybe defining what and how a negative base operates) and it could change a lot of equations and possibly some other areas of mathematics by simplifying complex equations. It would need to show that a complex value can be represented and manipulated as a single valued number, not the more cumbersome x iy representation involving an unknown placeholder. That would change both mathematics and physics in a fundamental way.
Best to you,
Don