Word meaning "to begin simple but to become complex"

If you're willing to accept an idiom, you might try spiral out of control.

"The project started off easy, but quickly spiraled out of my control."

Or you could possibly use the word exponentially to convey your meaning.

"The project started off easy, but grew exponentially harder as the work progressed."

How about 'snowballing complexity'.

If the project became complex because you decided you needed do more than you originally intended then consider scope creep.

Scope creep can apply to projects that weren’t necessarily simple to begin with, but definitely does mean that they have become much more complex than originally envisioned. In this case, the emphasis is less on “things were more complex than we gave them credit for,” and more on “no one seems to know when to cut off new ideas.”

From Wikipedia:

Scope creep (also called requirement creep, function creep and feature creep) in project management refers to uncontrolled changes or continuous growth in a project’s scope. This can occur when the scope of a project is not properly defined, documented, or controlled. It is generally considered harmful.

If you're looking for a single word, complexify [M-W] fits the bill:

My proposal would simplify the process, whereas yours would needlessly complexify it.